N^{o} d’ordre: 141-2005 Année 2005

THÈSE

présentée devant

L’UNIVERSITÉ CLAUDE BERNARD, LYON 1

pour l’obtention du

DIPLÔME DE DOCTORAT

(arrêté du 25 avril 2002)

présentée et soutenue publiquement le 23/09/2005 par

Jean-Baptiste GRAMAIN

GENERALIZED BLOCK THEORY

Au vu des rapports de:

M. Michel BROUÉ, M. Burkhard KÜLSHAMMER.

Devant le jury composé de:

M. Michel BROUÉ, M. Marc CABANES, M. Meinolf GECK, directeur de thèse, M. Burkhard KÜLSHAMMER, M. Olivier MATHIEU, M. Geoffrey R. ROBINSON, directeur de thèse.

2 À mon père.

Une civilisation sans la science, c’est aussi absurde qu’un poisson sans

bicyclette. Pierre Desproges, Je baisse.

It is not how fast you go that matters, it is the object of your journey. It is not how you send a message, it is what the value of the message may be.

Sir Arthur Conan Doyle, The land of mist.

Je sers la science et c’est ma joie. Basile (disciplus simplex).

4

Remerciements

Merci d’abord à Geoffrey Robinson qui, malgré la distance, a su diriger mes recherches, et dont le travail a inspiré le mien. Merci de m’avoir fait découvrir, parfois dans la douleur, les représentations de groupes, et de m’avoir donné envie de poursuivre dans ce domaine.

À mon autre directeur, Meinolf Geck, je veux exprimer ma profonde gratitude. Je lui suis extrêmement reconnaissant pour le temps qu’il a passé à répondre à toutes mes questions, même, et surtout, les plus stupides, à m’encourager et à me soutenir dans les moments difficiles, et à me conseiller sur la route à suivre. Je le remercie aussi pour son amitié, son humanité, et les bons moments passés ensemble. Merci également à son épouse Lacri pour sa gentillesse et son soutien.

Merci à Michel Broué et Burkhard Külshammer d’avoir accepté d’être rapporteurs pour cette thèse et de faire partie du jury. Merci également aux autres membres du jury: Marc Cabanes et Olivier Mathieu.

Je suis reconnaissant à l’Université de Birmingham (Angleterre) et à l’École Polytechnique Fédérale de Lausanne (Suisse), ainsi qu’à leurs personnels, de m’avoir accueilli pendant les quelques mois que j’y ai passés.

Je veux remercier du fond du coeur ma famille et mes ami(e)s qui, depuis des années, ont cru en moi (souvent plus que moi) et m’ont soutenu. Merci à tou(te)s d’avoir toujours été là, pour partager les joies et les peines, les doutes et les espoirs. Merci et bravo en particulier à ceux qui ont eu à me supporter au quotidien pendant ces trois années: mes parents, Lisa et Pauline. Merci aussi à celles et ceux qui m’ont accueilli et entouré quand je fuyais Lyon pour le soleil du Sud ou les montagnes.

Merci à tous mes compagnons d’aventures mathématiques: Sophie, Martin, Florence et Laurian, même si notre piscine de bureau n’aura jamais vu le jour, David, Séverine et Ammar, Daniel et Jean. Je remercie tout spécialement Attila, Nicolas et Olivier pour l’aide et l’amitié qu’ils m’ont données, chacun à leur manière, et toujours au bon moment.

Merci à tous les producteurs de café et de tabac sans qui cette thèse n’aurait jamais vu le jour. Merci également à tous les producteurs de bière, ainsi qu’à tous ceux qui en ont bu avec moi (et dont la liste serait trop longue), que ce soit pour parler maths, littérature, politique ou religion, pour jouer aux cartes, ou pour le simple plaisir d’être ensemble et de boire une bière.

Je présente mes excuses à tous les arbres abattus pour servir de support à mes succès et à mes échecs (beaucoup plus nombreux). Merci aux montagnes et aux déserts qui m’ont accueilli et m’ont permis de me ressourcer et de m’émerveiller. Enfin, je veux remercier de tout coeur les Beatles, Beethoven, Terry Pratchett et Ian Rankin, qui ont toujours été disponibles et présents avec moi.

Contents

Introduction 9

Introduction 13

Notations 17

1 Generalized Block Theory 19

1.1 Block Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.1 Modular systems . . . . . . . . . . . . . . . . . . . . . 19 1.1.2 Brauer characters . . . . . . . . . . . . . . . . . . . . . 21 1.1.3 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.1.4 Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.1.5 Brauer’s First and Second Main Theorems . . . . . . . 25

1.2 Generalized Block Theory . . . . . . . . . . . . . . . . . . . . 30 1.2.1 Generalized blocks . . . . . . . . . . . . . . . . . . . . 30 1.2.2 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2.3 Second Main Theorem property . . . . . . . . . . . . . 32 1.2.4 π-blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.3 Generalized Blocks for Symmetric Groups . . . . . . . . . . . 35 1.3.1 Sections, blocks . . . . . . . . . . . . . . . . . . . . . . 35 1.3.2 Nakayama Conjecture . . . . . . . . . . . . . . . . . . 36 1.3.3 Second Main Theorem property . . . . . . . . . . . . . 38

2 Generalized Perfect Isometries 41 2.1 p^{k}-blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 (Generalized) Perfect Isometries . . . . . . . . . . . . . . . . . 42 2.3 Special Unitary Groups . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . 44 2.3.2 Irreducible characters, principal blocks . . . . . . . . . 46 2.3.3 A generalized perfect isometry . . . . . . . . . . . . . . 50

2.4 Suzuki Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4.1 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . 52 2.4.2 Irreducible characters of B . . . . . . . . . . . . . . . . 53 2.4.3 A generalized perfect isometry . . . . . . . . . . . . . . 58

2.5 Ree Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5.1 The groups G, B and U . . . . . . . . . . . . . . . . . 59 2.5.2 Conjugacy classes of U . . . . . . . . . . . . . . . . . . 60 2.5.3 Irreducible characters of U . . . . . . . . . . . . . . . . 63 2.5.4 Irreducible characters of B = N_{G}(U) . . . . . . . . . . 68 2.5.5 A generalized perfect isometry . . . . . . . . . . . . . . 71

3 Cartan Group, Generalized Characters 73 3.1 Cartan Group, Factors . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Generalized Characters . . . . . . . . . . . . . . . . . . . . . . 74

3.2.1 Order in the Cartan group . . . . . . . . . . . . . . . . 74 3.2.2 First observations . . . . . . . . . . . . . . . . . . . . . 75 3.2.3 The prime case . . . . . . . . . . . . . . . . . . . . . . 78

3.3 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.1 Abelian groups . . . . . . . . . . . . . . . . . . . . . . 79 3.3.2 Special unitary groups, Suzuki groups, Ree groups … 81 3.3.3 Defect and order in the Cartan group . . . . . . . . . . 95

3.4 Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.1 Some facts about ℓ-blocks of the symmetric group … 97 3.4.2 Invariant factors of the Cartan matrix . . . . . . . . . 99 3.4.3 Orders in the Cartan group . . . . . . . . . . . . . . . 101

4 Finite General Linear Group 107

4.1 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.1.1 Rational canonical form . . . . . . . . . . . . . . . . . 108 4.1.2 The Jordan decomposition . . . . . . . . . . . . . . . . 109 4.1.3 Primary decomposition . . . . . . . . . . . . . . . . . . 110

4.2 (X , Y)-sections . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.3 (X , Y)-blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.3.1 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.3.2 Irreducible characters of G . . . . . . . . . . . . . . . . 114 4.3.3 Murnaghan-Nakayama Rule for unipotent characters . 115 4.3.4 Nakayama Conjecture for unipotent blocks . . . . . . . 118

4.4 Second Main Theorem property . . . . . . . . . . . . . . . . . 120 4.4.1 (X , Y)-blocks of centralizers . . . . . . . . . . . . . . . 120 4.4.2 Second Main Theorem property for combinatorial unipo-

tent (X , Y)-blocks . . . . . . . . . . . . . . . . . . . . 121

Introduction

La théorie des blocs a été introduite au milieu du 20ème siècle par R. Brauer. Si G est un groupe fini et si Irr(G) est l’ensemble de ses caractères complexes irréductibles, alors à tout nombre premier p est associée une partition de Irr(G) en p-blocs. Les blocs correspondent aux termes d’une décomposition en somme directe d’idéaux bilatères de l’algèbre de groupe sur un anneau de valuation discrète complet. La plupart des propriétés des p-blocs viennent des propriétés arithmétiques p-locales des caractères et sont obtenues grâce à une passerelle entre caractéristique 0 et caractéristique p. Mais les p-blocs peuvent aussi être obtenus élémentairement à partir du découpage de G en p-sections. Tout élément g de G a une unique écriture g = g_{p}g′_{p} = g′_{p}g_{p} où g_{p} est un p-élément et g′_{p} est p-régulier (i.e. l’ordre de g_{p} est une puissance de p et celui de g′_{p} est premier à p). Deux éléments appartiennent à la même section si les p-éléments associés sont conjugués. On note C l’ensemble des éléments p-réguliers de G, et on définit un produit intérieur tronqué par

< χ, ψ >_{C}= \frac{1}{|G|} ∑_{g∈C} χ(g)ψ(g^{−1}) pour tous χ, ψ ∈ Irr(G).

Deux caractères χ et ψ sont dits directement C-liés si < χ, ψ >_{C}≠ 0 (sinon, ils sont dits orthogonaux sur C). Celle relation est réflexive (car 1 ∈ C) et symétrique. En l’étendant par transitivité, on obtient une relation d’équivalence sur Irr(G) dont les classes sont les p-blocs. Une conséquence du Deuxième Théorème de Brauer est que, si deux caractères irréductibles sont orthogonaux sur C, alors ils sont orthogonaux sur chaque p-section (C est la p-section de 1).

L’idée développée par B. Külshammer, J. B. Olsson et G. R. Robinson dans [18] est de faire la même construction pour d’autres unions C de classes de conjugaison. On obtient ainsi une partition de Irr(G) en C-blocks. Le but de cette thèse est l’étude des propriétés de ces blocs généralisés. Dans [18], les auteurs ont défini des ℓ-sections et des ℓ-blocs pour les groupes symétriques, où ℓ ≥ 2 est un entier quelconque. Ils prennent pour C l’ensemble des éléments, dits ℓ-réguliers, dont aucun des cycles n’est de longueur divisible par ℓ. Les ℓ-blocs obtenus satisfont un analogue de la Conjecture de Nakayama : deux caractères irréductibles χ^{λ} et χ^{μ} du groupe symétrique S_{n} (λ et μ sont des partitions de n) appartiennent au même ℓ-bloc si et seulement si λ et μ ont le même ℓ-cœur. De plus, ils satisfont un analogue du Deuxième Théorème de Brauer.

La première partie de la thèse expose brièvement la construction et les propriétés des blocs dans le cas classique, puis introduit la généralisation ci-dessus et décrit le cas des groupes symétriques.

La deuxième partie traite d’isométries parfaites généralisées. Soit G un groupe fini, U un p-sous-groupe de Sylow de G, et B le normalisateur de U dans G. On note B_{0} (respectivement b_{0}) le p-bloc de G (respectivement de B) qui contient le caractère trivial. Une conjecture de M. Broué prévoit que, si U est abélien, alors il existe une isométrie parfaite entre B_{0} et b_{0} ( cf [2]). Une telle isométrie parfaite implique une relation étroite entre les anneaux de caractères associés à B_{0} et b_{0}. Si on prend pour G un groupe de Suzuki Sz(q), de Ree (de type G_{2}) Re(q) ou spécial unitaire SU(3, q^{2}), chacun en caractéristique p, alors les p-sous-groupes de Sylow de G ne sont pas abéliens. Il est connu (cf Cliff [8]) que, dans le cas des groupes de Suzuki, il n’y a pas d’isométrie parfaite entre B_{0} et b_{0}. Néanmoins, nous démontrons que, dans chacun de ces cas, si l’on prend pour C (resp. D) l’ensemble des éléments de G (resp. B) dont l’ordre n’est pas divisible par p^{2}, alors il existe une bijection I entre B_{0} et b_{0} et des signes {ε(χ), χ ∈ Irr(G)} tels que, pour tous χ, ψ ∈ B_{0}, on a < χ, ψ >_{C}=< ε(χ)I(χ), ε(ψ)I(ψ) >_{D}. On parle d’isométrie parfaite généralisée entre B_{0} et b_{0}. Ce résultat est moins fort que celui annoncé par la conjecture de Broué, mais il met cependant en évidence le lien fort entre les anneaux de caractères de B_{0} et b_{0}. En particulier, leurs groupes de Cartan sont les mêmes.

La troisième partie de la thèse traite notamment de ce groupe de Cartan. On suppose que C est une union fermée de classes de conjugaison de G (i.e. si x ∈ C et si y engendre le même sous-groupe de G que x, alors y ∈ C). Pour χ ∈ Irr(G), on définit une fonction centrale χ^{C} de G par χ^{C}(g) = χ(g) si g ∈ C et χ^{C}(g) = 0 si g ∈ G \ C. On définit le sous-Z-module R(C) de CIrr(G) engendré par {χ^{C}; χ ∈ Irr(G)}, et le sous-Z-module P(C) de R(C) des éléments de R(C) qui sont des caractères généralisés de G (i.e. P(C) = R(C) ∩ ZIrr(G)). Alors R(C) et P(C) ont le même rang sur Z. En particulier, R(C)/P(C) est un groupe abélien fini, le C-groupe de Cartan de G, et, pour tout χ ∈ Irr(G), il existe un entier d > 0 tel que dχ^{C} ∈ ZIrr(G). Le plus petit tel d est appelé ordre de χ.

Le groupe de Cartan et les ordres des caractères sont bien connus dans le cas où C est l’ensemble des éléments p-réguliers de G. Dans cette partie, nous donnons une description du groupe de Cartan et des ordres des caractères dans le cas où G est abélien et C est l’ensemble des éléments p^{k}-réguliers de G (i.e. dont l’ordre n’est pas divisible par p^{k}), où p est premier et k > 0. Nous établissons également une formule pour l’ordre des caractères des groupes symétriques, quand C est l’ensemble des éléments ℓ-réguliers. Il s’avère que cette formule est un ℓ-analogue de la Formule du Crochet pour calculer le degré d’un caractère irréductible. En particulier, elle montre que, si ℓ est une puissance d’un nombre premier p, alors le rapport de l’ordre d’un caractère et de la p-partie de son degré est un invariant de chaque ℓ-bloc. Nous démontrons que cette dernière propriété est vraie en dehors des groupes symétriques sous certaines hypothèses (fortes) sur les blocs considérés. C’est aussi le cas si G est dans l’une des familles étudiées dans la deuxième partie.

La quatrième partie est consacrée au groupe linéaire fini GL(n, q). Nous y donnons des définitions de sections et de blocs, basées sur la séparation en deux de l’ensemble des polynômes irréductibles qui divisent le polynôme minimal d’un élément. Cette séparation est faite en comparant le degré des polynômes à un entier quelconque d. Bien que nous définissions des blocs sur l’ensemble des caractères irréductibles, nous n’obtenons des propriétés que pour les caractères unipotents. Ceux-ci sont paramétrés par les partitions de n, et nous montrons que, si deux caractères unipotents appartiennent au même bloc unipotent, alors les partitions de n associées ont le même d-coeur. Certaines unions de ces blocs, appelés blocs combinatoires satisfont donc un analogue de la Conjecture de Nakayama. Nous montrons qu’ils satisfont également un analogue du Deuxième Théorème de Brauer.

12 INTRODUCTION

Introduction

The representation theory of finite groups appeared and was first developped by G. Frobenius at the very end of the 19th century. We will assume that the reader is familiar with its ideas and techniques, in particular the notion of groups character and the structures of character rings and character tables. One of the major improvement in this theory was the introduction by Brauer, in the 1950’s, of modular representation theory. The study of the representations of a finite group over a field of prime characteristic lead Brauer to develop the block theory of this group. This is the starting point of this thesis. If G is a finite group and Irr(G) is the set of irreducible complex characters of G, then to each prime p is associated a partition of Irr(G) into p-blocks. The blocks correspond to the summands in the decomposition into a direct sum of two-sided ideals of the group algebra of G over a complete discrete p-valuation ring. Most of the properties of p-blocks come from the p-local arithmetic properties of characters, and can be obtained through a bridge between characteristic 0 and characteristic p. This link is obtained via the introduction of p-modular systems. However, the p-blocks can also be obtained in an elementary way, based on the distinction between p-regular and p-singular elements of G. If we let C be the set of p-regular elements of G, we can consider the restriction to C of the ordinary scalar product on characters of G. For χ, ψ ∈ Irr(G), we let

< χ, ψ >_{C}= \frac{1}{|G|} ∑_{g∈C} χ(g)ψ(g^{−1}).

Then χ and ψ are said to be directly C-linked if < χ, ψ >_{C}≠ 0, and orthogonal across C otherwise. Then direct C-linking is a reflexive (since 1 ∈ C) and symmetric binary relation on Irr(G). Extending it by transitivity, we obtain an equivalence relation (called C-linking) on Irr(G) whose equivalence classes are the p-blocks.

The idea introduced by B. Külshammer, J. B. Olsson and G. R. Robinson in [18] is to do the same construction for other unions C of conjugacy classes of G. We therefore get a partition of Irr(G) into generalized blocks, or Cblocks. It is the purpose of this thesis to study some properties of these generalized blocks, and to define interesting blocks in some classes of groups. In [18], the authors have defined ℓ-blocks for the symmetric groups, where ℓ ≥ 2 is any integer. To obtain this, they take C to be the set of ℓ-regular elements, i.e. none of whose cycle has length divisible by ℓ. The ℓ-blocks obtained in this way satisfy an analogue of the Nakayama Conjecture: two irreducible characters χ^{λ} and χ^{μ} of the symmetric group S_{n} (where λ and μ are partitions of n) belong to the same ℓ-block if and only if λ and μ have the same ℓ-core.

In the first part of this thesis, we present shortly the construction and properties of the blocks in the classical case. In particular, we state Brauer’s First and Second Main Theorems. A consequence of the latter is that irreducible characters in distinct p-blocks, which are thus orthogonal across the p-section of 1, are in fact orthogonal across each p-section. The end of the part is devoted to the presentation of the ideas and results in [18]. We introduce the generalization we mentionned above, in particular the notion of (X , Y)-section, which generalizes p-sections. This allows us to define a Second Main Theorem property, which is an analogue of Brauer’s Second Main Theorem in this setting (depending on the set C of conjugacy classes we start with, this property may be satisfied or not by the C-blocks). We then describe the case of symmetric groups. In particular, the ℓ-blocks of symmetric groups satisfy the Second Main Theorem property.

The second part deals with generalized perfect isometries. Let G be a finite group, p a prime, U a Sylow p-subgroup of G, and B the normalizer of U in G. Let B_{0} and b_{0} be the principal p-blocks (i.e. containing the trivial character) of G and B respectively. Then a conjecture of M. Broué states that, if U is Abelian, then there should be a perfect isometry between B_{0} and b_{0} (cf [2]). Such a perfect isometry induces a close relationship between the character rings associated with B_{0} and b_{0}. Now, if G is a Suzuki group Sz(q), a Ree group (of type G_{2}) Re(q) or a special unitary group SU(3, q^{2}), each in characteristic p, then the Sylow p-subgroups of G are not Abelian. Furthermore, it is known (cf Cliff [8]) that, in the case of Suzuki groups, there is no perfect isometry between B_{0} and b_{0}. However, we show that, in each of these cases, if we take C and D to be the sets of elements of G and B respectively whose order is not divisible by p^{2}, then there exists a bijection I between B_{0} and b_{0} and signs {ε(χ), χ ∈ Irr(G)} such that, for any χ, ψ ∈ B_{0}, we have < χ, ψ >_{C}=< ε(χ)I(χ), ε(ψ)I(ψ) >_{D}. We refer to this as a generalized perfect isometry between B_{0} and b_{0}. This result is weaker than the one announced by Boué’s Conjecture, but still enlighten the strong link between the character rings of B_{0} and b_{0}. In particular, their Cartan groups are isomorphic.

The Cartan group is presented in the third part of the thesis. We suppose that C is a closed union of conjugacy classes of the finite group G (i.e. if x ∈ C and if y generates the same subgroup of G as x, then y ∈ C). For χ ∈ Irr(G), we define a class function χ^{C} of G by χ^{C}(g) = χ(g) if g ∈ C and χ^{C}(g) = 0 if g ∈ G \ C. We define the Z-submodule R(C) of CIrr(G) spanned by {χ^{C}; χ ∈ Irr(G)}, and the Z-submodule P(C) of R(C) consisting of those elements of R(C) which are generalized characters of G (i.e. P(C) = R(C) ∩ ZIrr(G)). Then R(C) and P(C) have the same Z-rank. In particular, R(C)/P(C) is a finite Abelian group, called the C-Cartan group of G, and, for each χ ∈ Irr(G), there exists an integer d > 0 such that dχ^{C} ∈ ZIrr(G). The smallest such d is called the order of χ.

The Cartan group and the orders of characters are well-known when C is the set of p-regular elements of G. In this part, we give a description of the Cartan group and of the orders of characters in the case where G is Abelian and C is the set of elements whose order is not divisible by p^{k}, where p is a prime and k > 0. We also obtain a formula for the orders of characters of the symmetric group, when C is the set of ℓ-regular elements. It turns out that this formula can be seen as an ℓ-analogue of the Hook-Length Formula for computing the degree of an irreducible character. In particular, it implies that, if ℓ is a power of a prime p, then the quotient of the order of a character by the p-part of its degree is an invariant of each ℓ-block. We show that this last property is true outside the symmetric groups under some (strong) hypothesis on the blocks we consider. It is also the case in the three families of groups we study in the second part.

The fourth part is devoted to the finite general linear group GL(n, q). We give definitions of sections and blocks, based on the splitting into two of the set of irreducible polynomials dividing the minimal polynomial of an element. This splitting is made by comparing the degrees of these irreducible polynomials with any given integer d. Although we define blocks for the whole set of irreducible characters of GL(n, q), we only obtain properties for the unipotent characters. These are labelled by the partitions of n, and we show that, if two unipotent characters belong to the same unipotent generalized block, then the partitions labelling them have the same d-core. Thus some unions of these blocks, called combinatorial blocks, satisfy an analogue of the Nakayama Conjecture, and we show that they also satisfy the Second Main Theorem property.

16 INTRODUCTION

Notations

Throughout this thesis, we will use the following notations:

N the natural integers (including 0). Z the ring of integers. Q, C the fields of rationals and complex numbers. F_{q} the field with q elements (q a prime power). (F_{q}, +) the additive group of F_{q}. F_{q}^{×} the multiplicative group of F_{q} (the non-zero elements of F_{q}). F_{q}[X] the ring of polynomials in one indeterminate with coefficients in F_{q}. δ(f) the degree of the polynomial f ∈ F_{q}[X]. n_{p} the p-part of n (n a positive integer, p a prime). n_{p′} the p′-part of n. gcd(m, n) the greatest common divisor of the integers m and n. lcm(m, n) the least common multiple of the (positive) integers m and n. δ_{xy} the Kronecker delta function on x and y. A ∐ B the disjoint union of the sets A and B. |A| the order of the finite set A. rk_{Z}(M) the rank over Z of a rational matrix M or of a Z-module M. V ⊕ W the direct sum of the vector spaces V and W. GL(V) the field of automorphisms of the vector space V over the ground field. Id or 1 the identity map or the identity matrix. I_{n} the n by n identity matrix. g|_{W} the restriction to the subspace W of V of an element g in GL(V). Min(g) the minimal polynomial of g ∈ GL(V) over the ground field. Char(g) the characteristic polynomial of g ∈ GL(V) over the ground field. C_{d} the cyclic group of order d. G × H the direct product of two (finite) groups G and H (written G ⊕ H if G

and H are additive groups).

G ⋊ H a semi-direct product of G by H. Z(G) the center of G. G′ the derived subgroup of G. H ≤ G H is a subgroup of G.

H ⊲ G H is a normal subgroup of G. [G : H] the index of the subgroup H in G. N_{G}(H) the normalizer of H in G. C_{G}(g) the centralizer in G of an element g of G. C_{G}(H) the centralizer in G of a subgroup H of G. Cl(G) the set of conjugacy classes of G. k(C) the number of conjugacy classes in a union C of conjugacy classes of G. Cl_{G}(g) the conjugacy class of g in G. o(g) the order of g in G. < g > the subgroup of G generated by g. g_{p} the p-part of g (p a prime). g_{p′} the p-regular part of g. G_{p} the set of p-elements of G. G_{p′} the set of p-regular elements of G. Syl_{p}(G) the set of Sylow p-subgroups of G. O_{p}(G) the largest normal p-subgroup of G. Irr(G) the set of irreducible complex characters of G. 1_{G} the trivial character of G. ZIrr(G) the ring of generalized characters of G. < , >_{G} the scalar product for complex class functions of G. Res_{H}^{G}(χ) or χ|_{H} the restriction to the subgroup H of the character χ of G. Ind_{H}^{G}(ψ) the character of G induced from the character ψ of the subgroup H. χ ⊗ χ′ the tensor product of the characters (of the same group) χ and χ′.

Part 1

Generalized Block Theory

In this part, we introduce the notions and objects we will use in this thesis. In a first place, we give a quick overview of the “ordinary” block theory, introduced by R. Brauer. Then we give the definitions and first properties of generalized block theory, as presented in [18], and finally describe the generalized blocks for the symmetric groups defined in this latter article.

1.1 Block Theory

In this section, we let G be a finite group. For all the results in this section, and unless specified otherwise, we refer to Navarro [20]. Most of the notions and results in block theory come from arithmetic constructions and properties. It is this approach we use to describe Brauer characters and, later, blocks, even though, when generalizing, we will use another characterization of blocks.

1.1.1 Modular systems

If k is any field, the group algebra kG is semi-simple if and only if k is of characteristic 0 or p prime to |G|. For any field k, we write Irr(kG) the set of characters of irreducible kG-modules. If kG is semi-simple and if, for every simple kG-module V, V is absolutely simple (i.e. End_{kG}(V) = k.Id), then the structure of kG is enlighted by Wedderburn’s theorem on semi-simple algebras: kG is a direct sum of matrix algebras. For example, this is the case when k is algebraically closed.

When we move to characteristic p dividing |G|, we lose the semi-simplicity of the group algebra, but we want to keep the “absolute simplicity property”. This will be the motivation for the introduction of large enough p-modular systems. We first observe :

Proposition 1.1. There exists a field K ⊂ C such that [K : Q] < ∞ and such that any finite-dimensional representation of CG can be realised over K.

A theorem of Brauer makes this more precise by stating that, if m is the exponent of G, then K = Q(e^{\frac{2iπ}{m}}) is sufficient.

Similarly, for any prime p, there exists a finite field k_{0} ⊂ \overline{F}_{p} such that any finite-dimensional irreducible representation of \overline{F}_{p}G can be realised over k_{0}.

Now, let K be a field, K ⊂ C, [K : Q] < ∞, and let A_{K} = A ∩ K be the ring of algebraic integers in K (where A is the subring of C of algebraic integers). Then K = F rac(A_{K}) and we have

Theorem 1.2. Let {0} ≠ P ⊲ A_{K} be a prime ideal, and

O = \{\frac{a}{b} ∈ K = F rac(A_{K}) | b ∉ P\}

be the localisation of A_{K} in P. Then O is a subring of K, a local ring (i.e. with a unique maximal ideal M). Moreover, O is principal, and k = O/M is a finite field of characteristic p > 0, where Z ∩ P = pZ. If M = (π) with π ∈ O, then the ideals of O are exactly the (π^{i})_{i≥1}, and ⋂_{i≥1}(π^{i}) = {0}.

Furthermore, we have K = F rac(O) and every finite-dimensional KGmodule can be realised over O. For this theorem, presented like this, we refer to the graduate course on modular representation theory given in Lyon by M. Geck in 2001-2002, and, for another presentation, we refer to [1].

Definition 1.3. If p is a prime, K is a subfield of C, [K : Q] < ∞, and k and O are as in the above theorem, then (K, O, k) is called a p-modular system for G.

If p| |G|, and without further information about k, not all simple \overline{F}_{p}Gmodules need to be absolutely simple. However, adding to K a suitable |G|- th root of unity, we can suppose that k_{0} ⊂ k, so that any finite dimensional irreducible \overline{F}_{p}G-module can be realised over k.

Definition 1.4. A p-modular system is said to be large enough for G if any V ∈ Irr(CG) can be realised over K and any M ∈ Irr(\overline{F}_{p}G) can be realised over k, and if K and k contain all |G|-th roots of unity.

We have just shown that such a system exists, so from now on we let p be a prime and (K, O, k) be a p-modular system large enough for G. In particular, any simple KG- (resp. kG-) module is absolutely simple (and thus KG and kG are split algebras).

The ring O is often taken by authors to be complete (for some p-adic norm). This allows the lifting of idempotents from kG to OG. However, if the p-modular system is big enough for G, then the group algebra KG is split semisimple, and kG is a split algebra. Then, by a result of Heller, the Krull-Schmidt-Azuyama Theorem holds for OG-lattices (cf Curtis-Reiner [9], Theorem 30.18), and, furthermore, idempotents can be lifted from kG to OG (cf [9], Ex. 6.16).

In the sequel, we will note z ⟼ z^{∗} the canonical surjection O ⟶ k, also called reduction modulo p.

1.1.2 Brauer characters

Now we would like to turn to characters of kG-modules. However, because of the characteristic p, we have to use something else than just traces. We will describe the Brauer characters of G, which take values in O, thus allowing us to connect them with the elements of Irr(KG).

We need first a result about roots of unity :

Proposition 1.5. Let U_{p′} = {x ∈ K^{×} | x^{n} = 1 for some n ≥ 1, with n| |G| and p ∤ n}. Then U_{p′} ≤ K^{×}, U_{p′} ≤ O^{×} and, if |G| = p^{a}m (a ≥ 0, p ∤ m), then the restriction modulo p

{x ∈ K^{×} | x^{m} = 1} ⊂ U_{p′} ⟶ {x^{∗} ∈ k^{×} | (x^{∗})^{m} = 1} ⊂ k^{×}

is a bijection.

Let G_{p′} be the set of p-regular elements of G. If M is a kG-module, associated to the representation ρ of dimension d, and g ∈ G is p-regular of order o(g), then the eigenvalues (ξ_{i})_{1≤i≤d} of ρ(g) are o(g)-th roots of unity, so are in k^{×}. Therefore, there are unique w_{i} ∈ U_{p′} such that w_{i}^{∗} = ξ_{i}, 1 ≤ i ≤ d. We define the Brauer character φ_{M} of M by

\begin{aligned}φ_{M}: & G_{p′} & ⟶ & A_{K} ⊂ O \\ & g & ⟼ & ∑_{i=1}^{d} w_{i} \end{aligned} .

Then φ_{M} is a class function of G_{p′}, and, writing χ the character afforded by ρ, we have

∀g ∈ G_{p′}, φ_{M}(g)^{∗} = χ(g).

Moreover, if M ≅_{kG} M′, then φ_{M} = φ_{M′}.

We say that φ_{M} is irreducible if ρ is irreducible, and let IBr_{p}(G) be the set of irreducible Brauer characters of G.

The next theorem illustrates the importance of the Brauer characters. We write CF (G) and CF (G_{p′}) the sets of complex class functions of G and G_{p′} respectively. For χ ∈ CF (G), we denote by χ̂ the restriction of χ to G_{p′} (then χ̂ ∈ CF (G_{p′})). We have:

Theorem 1.6. (i) The set IBr_{p}(G) is a C-basis for CF (G_{p′}). (ii) For all χ ∈ Irr(KG), there exist unique non-negative integers (d_{χφ})_{φ∈IBr_{p}(G)} such that

χ̂ = ∑_{φ∈IBr_{p}(G)} d_{χφ}φ.

Then (i) implies in particular that |IBr_{p}(G)| is the number of p-regular classes of G.

The (integral) matrix D = ((d_{χφ}))_{χ∈Irr(KG),φ∈IBr_{p}(G)} is called the decomposition matrix of G. It has (maximum) rank |IBr_{p}(G)| (in particular, each column of D has at least one non-zero entry). We deduce that, if p ∤ |G|, then IBr_{p}(G) = {χ̂, χ ∈ Irr(KG)}, and the elements of Irr(kG) are obtained by taking the reduction modulo p of the elements of Irr(KG).

We let C = D^{t}D, and call C the Cartan matrix of G. Then C = ((c_{φ,ψ}))_{φ,ψ∈IBr_{p}(G)} is a positive definite symmetric matrix with non-negative integer coefficients.

For each φ ∈ IBr_{p}(G), we define a class function Φ_{φ} of G via

Φ_{φ} = ∑_{χ∈Irr(KG)} d_{χφ}χ

(Φ_{φ} corresponds to the column of φ in the decomposition matrix). We call Φ_{φ} the principal indecomposable character associated to φ. We then have the following:

Theorem 1.7. (i) For each φ ∈ IBr_{p}(G), Φ_{φ} vanishes outside G_{p′}. (ii) For all φ, ψ ∈ IBr_{p}(G), we have

< Φ_{φ}, ψ >_{G_{p′}}:= \frac{1}{|G|} ∑_{g∈G_{p′}} Φ_{φ}(g^{−1})ψ(g) = δ_{φψ}.

(iii) The set {Φ_{φ}, φ ∈ IBr_{p}(G)} is a Z-basis for the Z-module ZIrr_{p′}(G) of generalized characters of G vanishing outside G_{p′}.

Note that, for φ ∈ IBr_{p}(G),

Φ̂_{φ} = ∑_{χ∈Irr(KG)} d_{χφ}χ̂ = ∑_{ψ∈IBr_{p}(G)} c_{φψ}ψ,

so that (i) and (ii) implie that, for φ, ψ ∈ IBr_{p}(G),

< Φ_{φ}, Φ_{ψ} >_{G}=< Φ_{φ}, Φ_{ψ} >_{G_{p′}}= c_{φψ}.

Together with (iii), this shows that the Cartan matrix corresponds to the generalized definition of Cartan matrix we will give later.

1.1.3 Blocks

We now turn to the definitions of p-blocks of Irr(KG) and IBr_{p}(G). To each χ ∈ Irr(KG) are associated a primitive idempotent e_{χ} of Z(KG) and an irreducible (linear) representation ω_{χ} of Z(KG) given by:

e_{χ} = \frac{χ(1)}{|G|} ∑_{g∈G} χ(g^{−1})g

and

ω_{χ}(C̃) = \frac{|C|χ(g_{C})}{χ(1)},

where C is any conjugacy class of G, with representative g_{C}, and C̃ is the sum of the elements in C.

Then, for all χ, χ′ ∈ Irr(KG), we have ω_{χ}(e_{χ′}) = δ_{χχ′}. Furthermore, for each χ ∈ Irr(KG), the restriction ω_{χ}^{∗} is an irreducible representation of Z(kG).

The p-blocks we define correspond to the decomposition of OG into a direct sum of ideals OGe, where e is a primitive idempotent of Z(OG). We define the Brauer graph of G: the vertices are the χ ∈ Irr(KG), and two characters χ ≠ χ′ ∈ Irr(KG) are linked if there exists φ ∈ IBr_{p}(G) such that d_{χφ} ≠ 0 ≠ d_{χ′φ}. We then have the following:

Theorem/Definition 1.8. (i) Two characters χ ≠ χ′ ∈ Irr(KG) are linked

in the Brauer graph if and only if \frac{1}{|G|} ∑_{g∈G_{p′}} χ(g)χ′(g^{−1}) ≠ 0.

(ii) An element e ∈ Z(OG) is a primitive idempotent of Z(OG) if and only if e = ∑_{χ∈B} e_{χ} for some connected component B of the Brauer graph. The set B ⊂ Irr(KG) is then called a p-block of Irr(KG) (or just of G). (iii) Two characters χ, χ′ ∈ Irr(KG) belong to the same p-block of G if and only if ω_{χ}^{∗} = ω_{χ′}^{∗}.

Note that the p-blocks do not depend on the p-modular system we choose (cf [20]). We let Bl(G) be the set of p-blocks of G. We have

Proposition 1.9. For B ∈ Bl(G), we let ω_{B} = ω_{χ}^{∗} for any χ ∈ B, and e_{B} = ∑_{χ∈B} e_{χ}. Then

(i) {ω_{B}, B ∈ Bl(G)} is the complete set of (distinct) irreducible representations of Z(kG). (ii) {e_{B}^{∗}, B ∈ Bl(G)} is the complete set of primitive idempotents of Z(kG). (iii) For all B, B′ ∈ Bl(G), we have ω_{B}(e_{B}^{∗}) = δ_{BB′}.

For B ∈ Bl(G), the set {φ ∈ IBr_{p}(G) | ∃χ ∈ B, d_{χφ} ≠ 0} is called a pblock of IBr_{p}(G), and written IBr_{p}(B). Up to reordering lines and columns, we see that the p-blocks of Irr(KG) and IBr_{p}(G) correspond to a diagonal block decomposition of the decomposition matrix, and thus of the Cartan matrix. For B ∈ Bl(G), we define in a natural way the decomposition matrix D_{B} and Cartan matrix C_{B} of B.

1.1.4 Defect

We now give the definitions of defect of a character, a block, or a conjugacy class. With these, we obtain informations on generalized characters and the invariant factors of the Cartan matrices. The results we mention give answers in the prime case to the questions we will study in the third part of this thesis.

Definition 1.10. For χ ∈ Irr(KG), we define the defect d(χ) of χ to be the exact power of p dividing the integer \frac{|G|}{χ(1)}, i.e.

p^{d(χ)} = |\frac{|G|}{χ(1)}|_{p} .

We then define the defect d(B) of a block B of Irr(KG) to be

d(B) = max_{χ∈B} d(χ).

For C a conjugacy class of G, we define the defect d(C) of C by

p^{d(C)} = |C_{G}(g)|_{p} for g ∈ C.

For B a block of G, we let n_{B} and r_{B} be the numbers of irreducible (complex) characters in B and irreducible Brauer characters in B respectively.

The case of blocks of defect 0 is described by the following Theorem 1.11. Let B be a p-block of G and χ ∈ B. Then the following are equivalent:

(i) d(B) = 0. (ii) n_{B} = r_{B}. (iii) d(χ) = 0. (iv) χ is the only irreducible character in B.

(v) χ vanishes on p-singular elements. (vi) χ(1)_{p} = |G|_{p}.

To prove the next theorem, one needs the following lemma, which is interesting in itself for our work:

Lemma 1.12. Let χ ∈ Irr(KG). We define a class function χ̃ of G by letting, for g ∈ G,

χ̃(g) = \{ \begin{matrix}χ(g) if g is p-regular \\ 0 if g is p-singular \end{matrix} .

Then p^{d(χ)}χ̃ is a generalized character, while p^{d(χ)−1}χ̃ is not.

We then have the following result on Cartan matrices:

Theorem 1.13. Let B be a p-block of G. Then

(i) The elementary divisors of the Cartan matrix C_{B} divide p^{d(B)}, and at least one of them is equal to p^{d(B)}. In particular, n_{B} > r_{B} unless d(B) = 0. (ii) C_{B} has exactly one elementary divisor equal to p^{d(B)}.

Finally, we mention the following:

Theorem 1.14. The elementary divisors of the Cartan matrix C of G are

{p^{d(C)} | C p-regular class of G}.

1.1.5 Brauer’s First and Second Main Theorems

Now, we want to state Brauer’s First and Second Main Theorems, which relate blocks of a group G and of certain subgroups of G. These theorems have many applications in representation theory. In order to state them, we first introduce the notion of defect group and the Brauer correspondence. For this presentation, we refer to Goldschmidt [14] and Navarro [20].

Defect groups

The idea of defect groups is to associate to each p-block of G a certain conjugacy class of p-groups of G.

For subgroups H, L, M of G, we will write H ⊂_{L} M (resp. H =_{L} M) if H^{l} ⊂ M (resp. H^{l} = M) for some l ∈ L.

For each conjugacy class C of G with g_{C} ∈ C, we define a defect group δ(C) of C to be a Sylow p-subgroup of C_{G}(g_{C}) (δ(C) is therefore determined up to g-conjugacy).

For B a p-block of G and C a conjugacy class of G, we let

a_{B}(C̃) = \frac{1}{|G|} ∑_{χ∈B} χ(1)χ(g_{C}^{−1}),

where g_{C} ∈ C and C̃ = ∑_{g∈C} g. Then e_{B} = ∑_{C} a_{B}(C̃)C̃, and a_{B}(C̃) = 0 if C is a p-singular class.

Proposition/Definition 1.15. If B is any p-block of G, then there exists a p-regular class C such that ω_{B}(C̃) ≠ 0 ≠ a_{B}(C̃)^{∗}. Such a class is called a defect class for B, and δ(C) is called a defect group for B.

Proposition 1.16. Let δ(B) be a defect group for the block B and C a conjugacy class of G. Then

(i) If ω_{B}(C̃) ≠ 0, then δ(B) ⊂_{G} δ(C). (ii) If a_{B}(C̃)^{∗} ≠ 0, then δ(C) ⊂_{G} δ(B). (iii) δ(B) is determined up to G-conjugation. (iv) |δ(B)| = p^{d(B)}.

Finally,

Proposition 1.17. If P is a normal p-subgroup of G and B is a block of G, then P ⊂_{G} δ(B).

The Brauer correspondence

Let H be a subgroup of G. If λ is any linear functional on Z(kH) and C is a conjugacy class of G, we define λ^{G}(C̃) = λ((C ∩ H)^{∼}) and extend λ^{G} linearly to a linear functional on Z(kG).

If λ = ω_{b} for some block b of H, then ω_{b}^{G} may or may not be an algebra homomorphism. If it is, then ω_{b}^{G} = ω_{B} for some B ∈ Bl(G); we say that b^{G} is defined and b^{G} = B. The correspondence b ⟼ b^{G} is called the Brauer correspondence.

Defect groups behave well under the Brauer correspondence :

Proposition 1.18. Suppose H is a subgroup of G with a block b for which b^{G} is defined. Then there exists a p-subgroup P of G such that

δ(b) ⊂_{H} P =_{G} δ(b^{G}).

We will obtain a sufficient condition for the existence of b^{G}. Suppose H, subgroup of G, is such that C_{G}(P) ⊲ H for some p-subgroup P of G. For any conjugacy class C of G, define μ(C̃) = (C ∩ C_{G}(P))^{∼}. Since C_{G}(P) ⊲ H, C ∩ C_{G}(P) is a union of conjugacy classes of H, and μ extends linearly to a map Z(kG) ⟶ Z(kH).

Proposition/Definition 1.19. The map μ: Z(kG) ⟶ Z(kH) is an algebra homomorphism, called the Brauer homomorphism.

Theorem 1.20. Suppose P is a p-subgroup of G and P C_{G}(P) ⊂ H ⊂ N_{G}(P), then b^{G} is defined for all blocks b of H, and ω_{b^{G}} = ω_{b} ∘ μ. Moreover, if B is a block of G, then B = b^{G} for some block b of H if and only if P ⊂_{G} δ(B), in which case

μ(e_{B}^{∗}) = ∑_{b^{G}=B} e_{b}^{∗}.

Brauer’s Theorems

We can now state Brauer’s First and Second Main Theorems.

Theorem 1.21. (Brauer’s First Main Theorem) Suppose P is a p-subgroup of G. Then the Brauer correspondence is a bijection between the set of blocks of N_{G}(P) with defect group P and the set of blocks of G with defect group P.

As a consequence, we mention the following theorem concerning principal blocks. The principal block is the one containing the trivial character. It is therefore of maximal defect, so any of its defect groups is a Sylow p-subgroup of G. The following theorem can also be seen as an immediate consequence of Brauer’s Third Main Theorem, as stated in [20].

Theorem 1.22. Suppose H is a subgroup of G, b is a block of H, and C_{G}(δ(b)) ⊂ H. Then b^{G} (which is defined) is the principal block of G if and only if b is the principal block of H.

In particular, if B_{0} is the principal block of G and δ(B_{0}) =_{G} P ∈ Syl_{p}(G), then B_{0} is the image under the Brauer correspondence of the principal block b_{0} of N_{G}(P). The idea of Broué’s Conjecture (cf Part 2) is that, if P is Abelian, there should be a deeper correspondence, at the level of characters, namely a perfect isometry, between B_{0} and b_{0}.

Before stating Brauer’s Second Main Theorem, we need the following:

Lemma 1.23. Let x be a p-element of G, and let H = C_{G}(x). For χ ∈ Irr(KG) and φ ∈ IBr_{p}(H), there exist unique d_{χφ}^{x} ∈ C such that

χ(xy) = ∑_{φ∈IBr_{p}(H)} d_{χφ}^{x}φ(y)

for all p-regular y ∈ H.

We have

d_{χφ}^{x} = ∑_{ψ∈Irr(KH)} \frac{< Res_{H}^{G}(χ), ψ >_{H} ψ(x)}{ψ(1)}d_{ψφ}

(where the d_{ψφ}’s are the “ordinary” decomposition numbers). The d_{χφ}^{x}’s are called the generalized decomposition numbers.

Theorem 1.24. (Brauer’s Second Main Theorem) Let x be a p-element of G and b be a p-block of C_{G}(x). Then, if χ ∈ Irr(G) is not in b^{G}, then d_{χφ}^{x} = 0 for each φ ∈ IBr_{p}(b).

(Note that, with the above hypotheses, b^{G} is always defined.)

Corollary 1.25. Let x be a p-element of G, and y a p-regular element of C_{G}(x). Suppose B is a block of G and χ ∈ B. Then

χ(xy) = ∑_{b∈Bl(C_{G}(x)), b^{G}=B} ∑_{μ∈IBr_{p}(b)} d_{χμ}^{x}μ(y).

Using the definitions of the d_{χμ}^{x}’s, then the definition of the d_{ψμ}’s and the fact that x is central in C_{G}(x) = H, this can be reformulated as:

Corollary 1.26. Suppose B is a block of G and χ ∈ B. Then, for any p-element x ∈ G and p-regular element y ∈ H = C_{G}(x), we have

χ(xy) = ∑_{b∈Bl(H), b^{G}=B} ∑_{ψ∈b} < Res_{H}^{G}(χ), ψ >_{H} ψ(xy).

If θ is a class function of G and B is a block of G, we let

θ^{B} = ∑_{χ∈B} < θ, χ >_{G} χ.

Then θ = ∑_{B∈Bl(G)} θ^{B}. Another useful consequence of Brauer’s Second Main Theorem is the following:

Theorem 1.27. Let θ be a class function of G and let x ∈ G be a p-element. Suppose θ(xy) = 0 for all p-regular y ∈ C_{G}(x). Then, for each block B of G and p-regular y ∈ C_{G}(x), we have θ^{B}(xy) = 0.

Let B be a block of G, χ ∈ B, and let x ∈ G be a p-element and H = C_{G}(x). For any block b of H, we write

χ^{(b)} = ∑_{ψ∈b} < Res_{H}^{G}(χ), ψ >_{H} ψ

(χ^{(b)} is a generalized character of H). Then Res_{H}^{G}(χ) = ∑_{b∈Bl(H)} χ^{(b)}, and, by Corollary 1.26

∑_{b∈Bl(H), b^{G}≠B} χ^{(b)}(xy) = 0

for all p-regular y ∈ H.

Applying Theorem 1.27 in H (in which x is central), we get that, if b ∈ Bl(H) and b^{G} ≠ B, then χ^{(b)}(xy) = 0 for all p-regular y ∈ H. Hence, if there exists a p-regular y ∈ H such that χ^{(b)}(xy) ≠ 0 for some b ∈ Bl(H), then b^{G} = B (see the Second Main Theorem property later).

Theorem 1.27 has an important corollary:

Corollary 1.28. (Block Orthogonality) Let g, h ∈ G be such that g_{p} and h_{p} are not conjugate in G. Then, for each p-block B of G, ∑_{χ∈B} χ(g^{−1})χ(h) = 0.

This has a very important consequence. Before stating it, we define the p-sections of G. For g ∈ G, the p-section of g in G is the set of elements of G whose p-part is conjugate to g_{p}. Then G is a disjoint union of p-sections. The p-section of 1 is just the set of p-regular elements of G.

Take g ∈ G, and write π the p-section of g in G. The last corollary can then be written:

∀B ∈ Bl(G), ∀h ∈ G \ π, ∑_{χ∈B} χ(g^{−1})χ(h) = 0.

We say that each block of G separates π from its complement.

If π is a p-section of G, we say that two class functions η and θ of G are orthogonal across π if

\frac{1}{|G|} ∑_{g∈π} θ(g)η(g^{−1}) = 0.

By construction, we have seen that, if two irreducible characters of G belong to distinct p-blocks, then they are orthogonal across p-regular elements. We have in fact much better:

Theorem 1.29. If two irreducible characters of G belong to distinct p-blocks, then they are orthogonal across each p-section of G.

(This can be proved by using Corollary and adapting the argument given in [18] after Corollary 1.2)

1.2 Generalized Block Theory

In this section, we introduce the notion of generalized blocks, as introduced in [18]. The basis for “ordinary” block theory was the separation of a group between p-regular and p-singular elements. Now the idea is to use another set of conjugacy classes to split the group. This leads to a new definition of block, which we present in a first paragraph. We then give a definition of sections to partition the group, and state what an analogue of Brauer’s Second Main Theorem could look like with this definition (depending on the set of conjugacy classes we start with, the generalized blocks we obtain may or may not satisfy this property). For all the definitions and results in sections 1.2.1, 1.2.2 and 1.2.3, we refer to Külshammer, Olsson and Robinson [18]. We then give in section 1.2.4 an overview of the first natural generalization, the case of π-blocks.

1.2.1 Generalized blocks

Let G be a finite group, and C be a union of conjugacy classes of G. We say that C is closed if the following is true: for any x ∈ C, whenever y is an element of G which generates the same subgroup as x, then y ∈ C. The notion of closed set of conjugacy classes was introduced by M. Suzuki in [24]. Now let us fix a closed set C of conjugacy classes of G, and assume that 1 ∈ C. We let C′ = G \ C. Let Irr(G) be the set of complex irreducible characters of G. For any complex class function φ of G, we define φ^{C} to be the class function of G which agrees with φ on C and vanishes outside C.

Two characters χ, ψ ∈ Irr(G) are said to be directly C-linked if

< χ, ψ >_{C}:= \frac{1}{|G|} ∑_{x∈C} χ(x)\overline{ψ(x)} ≠ 0.

If < χ, ψ >_{C}= 0, then χ and ψ are said to be orthogonal across C. We call < χ, ψ >_{C} the C-contribution associated to χ and ψ. Note that < χ, ψ >_{C}=< χ^{C}, ψ >_{G}=< χ, ψ^{C} >_{G}=< χ^{C}, ψ^{C} >_{G}. Direct C-linking is a symmetric and reflexive (since 1 ∈ C) binary relation on Irr(G). Extending it by transitivity to an equivalence relation (called C-linking), we obtain a partition of Irr(G) into C-blocks. We refer to C-blocks of Irr(G) as C-blocks of G. Two characters χ, ψ ∈ Irr(G) thus belong to the same C-block of G if there exists a sequence χ_{0} = χ, χ_{1}, …, χ_{r} = ψ ∈ Irr(G) such that, for each 0 ≤ i ≤ r − 1, the characters χ_{i} and χ_{i+1} are directly C-linked. We say that χ and ψ are Clinked.

Note that if we take C to be the set of p-regular elements of G (p a prime), then the C-blocks of G are just its p-blocks.

We have the following:

Proposition 1.30. If g ∈ C and h ∈ C′, then, for each C-block B of G,

∑_{χ∈B} χ(g^{−1})χ(h) = 0.

Hence C-blocks separate C from C′.

The matrix Γ(C, G) = ((< χ, ψ >_{C}))_{χ,ψ∈Irr(G)} is called the C-contribution matrix of G. It has (if we list the elements of Irr(G) block by block) a diagonal block decomposition corresponding to the C-blocks of G. If B is a C-block of G, we let Γ(C, B) = ((< χ, ψ >_{C}))_{χ,ψ∈B} be the C-contribution matrix of B.

Using Galois Theory, and since C is closed, we see that the C-contributions are rationals.

1.2.2 Sections

In the same way we can partition G into p-sections, we would like a more general definition of sections, with the same properties, and which we can adapt to our situation. This includes relating these sections to some C-blocks as defined above, and defining blocks for the centralizers of certain elements (corresponding to the p-elements in the p-case).

Let X be a union of conjugacy classes of G containing the identity. Suppose that, for each x ∈ X, there is a union Y(x) of conjugacy classes of C_{G}(x) such that:

(i) 1 ∈ Y(x), (ii) Two elements of xY(x) are G-conjugate if and only if they are C_{G}(x)- conjugate, (iii) C_{G}(xy) ≤ C_{G}(x) for each y ∈ Y(x). Suppose also that Y(x^{g}) = Y(x)^{g} for all x ∈ X and g ∈ G, and that G is the disjoint union of the conjugacy classes (xy)^{G}, as x runs through a set of representatives for the G-conjugacy classes in X and y runs through a set of representatives for the C_{G}(x)-conjugacy classes in Y(x).

For example, if we take X to be the set of p-elements of G, then, for each x ∈ X, Y(x) can be taken to be the set of p-regular elements of C_{G}(x).

For any x ∈ X, we call the union of the G-conjugacy classes meeting xY(x) the Y-section of x. The hypotheses we made ensure that, for each x ∈ X, induction of complex class functions gives an isometry from the space of class functions of C_{G}(x) vanishing outside xY(x) onto the space of class functions of G vaniching outside the Y-section of x.

We define an (X , Y)-block of G as a Y(1)-block of G in the sense we defined above. For x ∈ X, we define the (X , Y)-blocks of C_{G}(x) to be the smallest (non-empty) subsets of Irr(C_{G}(x)) such that irreducible characters in distinct subsets are orthogonal across xY(x).

Note that, because of Proposition 1.30, (X , Y)-blocks of G separate Y(1) from its complement. For x ∈ X, we can equally define (X , Y)-blocks of C_{G}(x) to be non-empty subsets of Irr(C_{G}(x)) which are minimal subject to separating Y(x) from its complement in C_{G}(x) (or, equivalently, to separating xY(x) from its complement (since x is central in C_{G}(x))).

1.2.3 Second Main Theorem property

Suppose we have sets X ⊂ G and Y(x) for x ∈ X as above. Suppose χ ∈ Irr(G) and β is a union of (X , Y)-blocks of C_{G}(x) for some x ∈ X. We define a generalized character χ^{(β)} of C_{G}(x) via

χ^{(β)} = ∑_{μ∈β} < Res_{C_{G}(x)}^{G}(χ), μ > μ.

Definition 1.31. Let x ∈ X and b be an (X , Y)-block of C_{G}(x). We say that an (X , Y)-block B of G dominates b if there exist χ ∈ B and y ∈ Y(x) such that χ^{(b)}(xy) ≠ 0.

We see that, for x ∈ X, if χ ∈ B for some (X , Y)-block B of G, then, for each y ∈ Y(x), we have χ(xy) = ∑_{b} χ^{(b)}(xy), where b runs through the set of (X , Y)-blocks of C_{G}(x) dominated by B.

Note that, for x ∈ X, each (X , Y)-block of C_{G}(x) is dominated by at least one (X , Y)-block of G.

Definition 1.32. We say that the (X , Y)-blocks of G satisfy the Second Main Theorem property if, for each x ∈ X and each (X , Y)-block b of C_{G}(x), b is dominated by a unique (X , Y)-block of G.

As we have remarked in the previous section, if, for some prime p, X is the set of p-elements of G and if, for each x ∈ X, Y(x) is the set of p-regular elements of C_{G}(x), then the (X , Y)-blocks of G (i.e. the p-blocks of G) have the Second Main Theorem property.

Using the fact that, for x ∈ X, irreducible characters of C_{G}(x) in distinct (X , Y)-blocks are orthogonal across xY(x), one proves easily the following:

Proposition 1.33. The (X , Y)-blocks of G satisfy the Second Main Theorem property if and only if, for each (X , Y)-block B of G, there is, for each x ∈ X, a (possibly empty) union of (X , Y)-blocks β(x, B) of C_{G}(x) such that, for each irreducible character χ ∈ B and each character μ ∈ β(x, B), we may find a complex number c_{χ,μ} such that, for each y ∈ Y(x), we have

χ(xy) = ∑_{μ∈β(x,B)} c_{χ,μ}μ(xy),

and, furthermore, β(x, B) and β(x, B′) are disjoint whenever B and B′ are distinct (X , Y)-blocks of G.

The following theorem enlighten the link between the Second Main Theorem property and Brauer’s Second Main Theorem:

Theorem 1.34. Suppose that the (X , Y)-blocks of G satisfy the Second Main Theorem property. Then:

(i) Irreducible characters of G which are in distinct (X , Y)-blocks are orthogonal across each Y-section of G. (ii) If x ∈ X and ∑_{χ∈Irr(G)} a_{χ}χ is a class function which vanishes identically on the Y-section of x in G, then, for each (X , Y)-block B of G, ∑_{χ∈B} a_{χ}χ also vanishes identically on the Y-section of x in G. (iii) (X , Y)-blocks of G separate Y-sections of G.

1.2.4 π-blocks

One of the main problems we meet when moving from p-blocks to C-blocks is that we generally lose all the arithmetic on which the modular representation theory is based. In particular, the discrete valuation ring O has no analogue in general, and C-blocks are not related to idempotents of a group algebra.

One case in which arithmetic arguments can still be used is that of π-blocks, where π is a set of primes. These were initially introduced and studied by Brauer (even though he didn’t publish his results), and then considered by many authors. From now on, we let G be a finite group and π be a set of primes. Different, but equivalent, definitions of π-blocks can be given. The first might be the more natural.

Definition 1.35. (Staszewski [23], definition 1.1) A non-empty subset B of Irr(G) is called a π-block of G if, for every p ∈ π, B is a union of p-blocks, and if B is minimal for this property.

Note that, if π = {p}, then a π-block is just a p-block.

Now the π-blocks can also be seen in terms of idempotents. For this definition, we refer to Staszewski [23] and Robinson [21]. Let ω = e^{2πi/|G|}. For any p ∈ π, we let O_{p} = \{\frac{α}{β} | α, β ∈ Z[ω] and, for any ideal ρ of Z[ω] containing p, β ∉ ρ\} be the p-adic integer ring of Q[ω] (with relative residue field F_{p}). We set O_{π} = ⋂_{p∈π} O_{p}. Note that O_{π} is no longer a local ring, but that it has only finitely many prime ideals. Then we have

Definition 1.36. A non-empty subset B of Irr(G) is a π-block of G if

\frac{1}{|G|} ∑_{χ∈B} ∑_{g∈G} χ(1)χ(g^{−1})g

is a primitive idempotent of Z(O_{π}G).

It can be shown (cf [21], Lemma 2) that π-blocks are, with the terminology we introduced before, C-blocks, where C is the set of π-regular elements of G (π-blocks are subsets of Irr(G) which are minimal subject to being orthogonal across π-regular elements). We also see that the π-sections of G satisfy the properties for (X , Y)-sections, X being the set of π-elements and Y(x) being the set of π-regular elements of C_{G}(x) for x ∈ X. In this setting, an (X , Y)- block of G is a π-block of G, and, for x ∈ X, an (X , Y)-block of C_{G}(x) is a π-block of C_{G}(x).

The π-blocks of a finite group have properties very similar to those of p-blocks. In particular, it is possible to define analogues of the principal indecomposable characters, the Brauer characters, the decomposition numbers and Cartan numbers, the corresponding decomposition matrix and Cartan matrix having the same kind of block structure as in the p-case (cf [23]). Staszewski also studies the defect of a π-block, and determines the elementary divisors of the Cartan matrices of G and of any given π-block of G.

The generalization of the notion of defect group appears to be more tricky. However, this can be done if G has a nilpotent Hall π-subgroup, and defect groups are unique up to conjugacy if G has furthermore a normal π-complement (cf [23]).

It is shown in Robinson [21] that π-blocks satisfy an analogue of Brauer’s Second Main Theorem. In particular, π-blocks separate π-sections, and irreducible characters in distinct π-blocks are orthogonal across each π-section (cf [21], Corollary 8). Robinson and Staszewski also show in [22] that, if G has a cyclic Hall π-subgroup, then the π-blocks of G satisfy an analogue of Brauer’s Third Main Theorem (cf [22], Theorem 2.1).

Finally, Broué and Michel have studied the π-blocks of finite reductive groups in [3]. Using the Deligne-Lusztig Theory, the irreducible (complex) characters of such a group G^{F} are partitioned into geometric conjugacy classes E(G^{F} , (s)), labelled by the conjugacy classes of semisimple elements s in the dual group G^{∗F^{∗}}. Broué and Michel show in particular that, if the defining characteristic of G^{F} does not belong to π, then the union E_{π}(G^{F} , (s)) of such geometric conjugacy classes for the parameter describing a π′-section, i.e. E_{π}(G^{F} , (s)) = ⋃_{x∈C_{G^{∗}}(s)^{F^{∗}}} E(G^{F} , (sx)) for some semisimple π′-element s of G^{∗F^{∗}}, is a union of π-blocks (cf [3], Theorem 2.2).

1.3 Generalized Blocks for Symmetric Groups

In this section, we give an overview of some of the results obtained in [18]. In this article, the authors define ℓ-blocks for the symmetric group S_{n}, where ℓ ≥ 2 is an arbitrary integer. These blocks can be related to two different definitions of (X , Y)-sections, which give distinct (X , Y)-blocks for the centralizers, but the same (X , Y)-blocks for S_{n}. The authors show that the ℓblocks of S_{n} thus defined satisfy an ℓ-analogue of the Nakayama Conjecture, and that they satisfy the Second Main Theorem property.

In all this section, we consider the symmetric group S_{n} for some n ≥ 1, and let ℓ ≥ 2 be an integer. We let π be the set of primes dividing ℓ.

1.3.1 Sections, blocks

Recall each element of a symmetric group can be written uniquely as a product of disjoint cycles.

Definition 1.37. An element of a symmetric group is said to be:

• an ℓ-cycle element if all its non-trivial cycles have length divisible by ℓ;

• ℓ-regular if it has no cycle of length divisible by ℓ (and ℓ-singular otherwise);

• an ℓ-element if it is an ℓ-cycle element such that each non-trivial cycle has length dividing a power of ℓ;

• π-regular if its order is not divisible by any prime in π.

If two elements x, y ∈ S_{n} are disjoint (i.e. each one fixing the points moved by the other), we write x ∗ y for the product xy. We have:

Proposition/Definition 1.38. Each element z ∈ S_{n} has unique factorizations:

z = x ∗ y = rs = sr

where x is an ℓ-cycle element, y is ℓ-regular, r is the π-part of x (thus r is an ℓ-element) and s is ℓ-regular. Any element commuting with z commutes with each of x, y, r and s (in particular, these elements all commute with each other). We call x the ℓ-cycle part of z, and r the ℓ-part of z.

Two elements of S_{n} are said to belong to the same ℓ-cycle section (respectively ℓ-section) of S_{n} if their ℓ-cycle parts (respectively ℓ-parts) are conjugate in S_{n}. Then each ℓ-section of S_{n} is a union of ℓ-cycle sections. The ℓ-section of 1 is the set of ℓ-regular elements of S_{n}, and is also an ℓ-cycle section.

We now define (X , Y)-blocks for S_{n}. We let X be the set of ℓ-elements of S_{n}. For each r ∈ X, the centralizer C = C_{S_{n}}(r) of r can be written C = C_{0} × C_{1}, where C_{1} is the pointwise stabilizer of the points moved by r (and thus a symmetric group), and C_{0} is the pointwise stabilizer of the points fixed by r. Then r ∈ C_{0}. We let Y(r) be the set of elements s_{0} ∗ s_{1} ∈ C_{0} × C_{1} such that s_{0} is a π′-element of C_{0} and s_{1} is an ℓ-regular element of C_{1}.

We let X′ be the set of ℓ-cycle elements of S_{n}. For each x ∈ X′, we let Y′(x) be the set of ℓ-regular elements of S_{n} which are disjoint from x.

Then the Y-sections of ℓ-elements are the ℓ-sections of S_{n}, and the Y′- sections of ℓ-cycle elements are the ℓ-cycle sections of S_{n}.

Note that Y(1) and Y′(1) are both equal to the set of ℓ-regular elements of S_{n}, so that the (X , Y)-blocks and the (X′, Y′)-blocks of S_{n} coincide, and could be defined by linking across ℓ-regular elements. We call them ℓ-blocks of S_{n}. An (X , Y)-block of the centralizer C of an ℓ-element is called an ℓblock of C, and an (X′, Y′)-block of the centralizer C of an ℓ-cycle element is called an ℓ-cycle block of C.

1.3.2 Nakayama Conjecture

The irreducible (complex) characters and conjugacy classes of S_{n} are labelled canonically by the partitions of n. For λ a partition of n (which we write λ ⊢ n), we let χ_{λ} be the irreducible character of S_{n} labelled by λ.

Given λ ⊢ n and any integer d, we obtain the d-core γ_{λ} of λ by removing from (the Young diagram of) λ all the d-hooks. We have the following (cf James and Kerber [17])

Theorem 1.39. (Nakayama Conjecture) Let p be a prime. Then two irreducible characters χ_{λ} and χ_{μ} of S_{n} belong to the same p-block if and only if λ and μ have the same p-core.

It is proved in [18] that the ℓ-blocks of S_{n} satisfy an ℓ-anologue of this theorem. We will give a sketch of the proof of this. The basis to start is the following (cf [17])

Theorem 1.40. (Murnaghan-Nakayama Formula) Let n = l + m, and let ρ ∗ σ ∈ S_{n}, where ρ is an m-cycle and σ is a permutation on the remaining l symbols. Let λ be a partition of n. Then

χ_{λ}(ρ ∗ σ) = ∑_{μ}(−1)^{L_{λμ}}χ_{μ}(σ),

where μ runs over all partitions μ of l which can be obtained from λ by deleting an m-hook, and L_{λμ} is the leg length of the deleted hook.

Now take an ℓ-cycle element x of S_{n}. Suppose that the cycle type of x is (ℓh_{1}, …, ℓh_{t}) (where we omit cycles of length 1). We call ρ = (h_{1}, …, h_{t}) the ℓ-type of x (and of x∗y if y is ℓ-regular). Then, if y is any element disjoint from x and λ is a partition of n, repeated use of the Murnaghan-Nakayama Formula gives

χ_{λ}(x ∗ y) = ∑_{μ⊢n−vℓ} m_{λμ}^{ρ}χ_{μ}(y), (†)

where v = h_{1} + ⋯ + h_{t}, and the coefficients m_{λμ}^{ρ} are integers. The coefficient m_{λμ}^{ρ} corresponds in some way to the set of paths in the lattice of partitions obtained by removing first an ℓh_{1}-hook, then an ℓh_{2}-hook, and so on, to obtain μ from λ. Now one can show that the removal of an ℓh-hook can be obtained by removing a sequence of h ℓ-hooks, and this implies that, if m_{λμ}^{ρ} ≠ 0, then λ and μ have the same ℓ-core.

Now, each ℓ-cycle section is characterized by the (common) ℓ-type of its elements, and (†) allows us to compute the contribution of two irreducible characters of S_{n} across any ℓ-cycle section. Then an induction argument shows that, if χ_{λ} and χ_{λ′} are (directly) linked across any ℓ-cycle section, then λ and λ′ have the same ℓ-core. This proves one implication of the Nakayama Conjecture:

Theorem 1.41. If χ_{λ}, χ_{λ′} ∈ Irr(S_{n}) belong to the same ℓ-block of S_{n}, then λ and λ′ have the same ℓ-core.

This implies that, for a given ℓ-core γ, the set B_{γ} = {χ_{λ} ∈ Irr(S_{n}) | γ_{λ} = γ} is a union of ℓ-blocks. We want to prove that B_{γ} is in fact a single ℓ-block. For this, we refer to Maróti [19]. For χ_{λ} ∈ B_{γ}, we let w be the number of ℓ-hooks which must be removed from λ to get γ_{λ}. Then w is independant on χ_{λ} ∈ B_{γ}, and is called the ℓ-weight of B_{γ} (and of any λ such that χ_{λ} ∈ B_{γ}).

Note that, if w = 0, then B_{γ} consists of a single character χ_{λ} (and λ = γ), and thus is trivially a single ℓ-block of S_{n}. We therefore fix an ℓ-core γ and suppose that w ≥ 1. The main ingredient of the proof is that there exists a generalized perfect isometry (this will be defined in Part 2) between B_{γ} and the set of irreducible characters of the wreath product Z_{ℓ} ≀ S_{w}. We will give some details about the wreath product and the generalized perfect isometry in Part 3. At this point, we only want to give some ideas of how the proof goes.

The first point is that there exists a canonically defined bijection Q between B_{γ} and Irr(Z_{ℓ} ≀ S_{w}) (we write Q for “quotient”, as will be seen in Part 3). The second ingredient is that there is an analogue of the Murnaghan-Nakayama Formula in Z_{ℓ}≀S_{w}. As in the case of symmetric groups, this allows us to relate the contribution of two characters across some set of elements of Z_{ℓ} ≀ S_{w} to the contributions of characters of a smaller wreath product across a smaller set of elements, and thus to build an induction argument. Comparing the results in S_{n} and Z_{ℓ} ≀ S_{w}, Maróti then shows that the contribution of two characters χ_{λ} and χ_{μ} in B_{γ} across ℓ-regular elements is equal, up to a sign, to the contribution of Q(χ_{λ}) and Q(χ_{μ}) across the so-called regular elements of Z_{ℓ} ≀ S_{w}. This gives us a very powerful bridge between Z_{ℓ} ≀ S_{w} and S_{n}. To conclude the proof, one proves that, in Z_{ℓ} ≀ S_{w}, every irreducible character is directly linked across regular elements to the trivial character. Using our bridge, this implies that any two characters in B_{γ} are both directly linked across ℓ-regular elements to a third character (whose image under Q is the trivial character of Z_{ℓ} ≀ S_{w}), and thus belong to the same ℓ-block of S_{n}. Finally, we have:

Theorem 1.42. (Generalized Nakayama Conjecture) Two irreducible characters χ_{λ} and χ_{λ′} of S_{n} belong to the same ℓ-block if and only if λ and λ′ have the same ℓ-core.

1.3.3 Second Main Theorem property

It is shown in [18] that the ℓ-blocks of S_{n} satisfy the Second Main Theorem property. It is only at this point, when considering centralizers, that the difference between (X , Y)-blocks and (X′, Y′)-blocks becomes apparent. We start with (X′, Y′)-blocks, which are somewhat easier to manipulate. This is why only the case of (X , Y)-blocks is presented in [18].

Take x ∈ X′. With the notations we used above, if x has ℓ-type ρ, where ρ is a partition of v, then we can see x as an element of S_{ℓv}. Then the centralizer of x is C_{S_{n}}(x) = C_{0} × C_{1}, where C_{0} = C_{S_{ℓv}}(x) and C_{1} = S_{n−ℓv}, and Y′(x) is the set of ℓ-regular elements of C_{1}. If we take μ_{0} ⊗μ_{1} and ν_{0} ⊗ν_{1} in Irr(C_{0} ×C_{1}), it is then easy to see, since x is central in C_{0}, that μ_{0} ⊗μ_{1} and ν_{0} ⊗ ν_{1} are directly linked across xY′(x) if and only if μ_{1} and ν_{1} are directly linked across ℓ-regular elements of C_{1} = S_{n−ℓv}. Thus the (X′, Y′)-blocks of C_{S_{n}}(x) are the Irr(C_{0}) ⊗ b’s, where b runs through the ℓ-blocks of C_{1}.

Now, for any (X′, Y′)-block B of S_{n}, labelled by the ℓ-core γ, we set β(x, B) = Irr(C_{0}) ⊗ b, where b is the ℓ-block of C_{1} labelled by γ, and, for any χ ∈ B and μ_{0} ⊗ μ_{1} ∈ Irr(C_{0}) ⊗ b, we set

c_{χ,μ_{0}⊗μ_{1}} = \{ \begin{matrix}m_{χμ_{1}}^{ρ} if μ_{0} = 1_{C_{0}} \\ 0 otherwise \end{matrix} ,

where the m_{χμ_{1}}^{ρ} are the coefficients appearing in formula (†). It is then clear, by formula (†), that, for each x ∈ X′, the β(x, B)’s and c_{χ,μ_{0}⊗μ_{1}}’s thus defined satisfy the hypotheses of Proposition 1.33. Hence the (X′, Y′)-blocks of S_{n} satisfy the Second Main Theorem property.

The case of (X , Y)-blocks is a bit more tricky. However, the main ingredients are the same. It can be shown that, for r ∈ X, writing C_{S_{n}}(r) = C_{0} ×C_{1} as indicated when we defined (X , Y)-sections, each ℓ-block of C_{S_{n}}(r) has the form Irr(C_{0}) × b for some ℓ-block b of C_{1}. The difference with the previous case is that, given s_{0} ∗ s_{1} ∈ Y(r), we can consider formula (†) applied to x = rs_{0} and y = s_{1}, but the coefficients appearing on the right side may depend on s_{0} (since the ℓ-types of r and rs_{0} may differ). However, even in this case, formula (†) can be used to show that, if an (X , Y)-block B of S_{n} dominates the (X , Y)-block Irr(C_{0}) ⊗ b of C_{S_{n}}(r), then B and b must be labelled by the same ℓ-core (cf [18]). Finally, we have

Theorem 1.43. The (X , Y)-blocks and the (X′, Y′)-blocks of S_{n} satisfy the Second Main Theorem property.

40 PART 1. GENERALIZED BLOCK THEORY

Generalized Perfect Isometries

In [18], the authors give a definition of generalized perfect isometry between (unions of) generalized blocks of two groups, and use it in the case of symmetric groups to prove that the ℓ-blocks of S_{n} satisfy an analogue of the Nakayama Conjecture (cf Part 1). In this part, we investigate some families of groups of Lie rank one (namely SU(3, q^{2}), Sz(q) and Re(q)), exhibiting generalized perfect isometries where there are none in the sense given originally by M. Broué, and thus proving a (weaker) analogue of one of Broué’s Conjectures in these cases.

2.1 p^{k}-blocks

We start by observing that some generalized blocks (namely p^{k}-blocks) can always be defined in a finite group. We take G any finite group, p a prime, and k ≥ 1 an integer. An element of G is said to be p^{k}-regular if its order is not divisible by p^{k}, and p^{k}-singular otherwise. We let C_{k} be the set of p^{k}-regular elements of G. Then C_{k} is a closed set of conjugacy classes of G, and, using the definitions of the previous part, we can define the C_{k}-blocks of G. We call them the p^{k}-blocks of G.

Take g =∈ G. We define the p^{k}-section of g to be its p-section if g is p^{k}-singular, and the set C_{k} of p^{k}-regular elements of G if g is p^{k}-regular. Then each p^{k}-section of G is a union a p-section. This is clear for p^{k}-singular sections, for they are already p-sections, and C_{k} is the union of the sections of elements of G whose p-part has order 1 or p.

Now the p-blocks of G satisfy the Second Main Theorem property (by Brauer’s Second Main Theorem), so that irreducible characters in distinct p-blocks of G are orthogonal across each p-section, and thus across each p^{k}- section. In particular, if two irreducible characters of G are directly C_{k}-linked, then they belong to the same p-block of G. This proves that any p-block of G is a union of p^{k}-blocks.

In the families of groups we will study in this part, we will be interested in p^{2}-blocks, where p is the defining characteristic.

2.2 (Generalized) Perfect Isometries

We find in [18] a definition of perfect isometry which generalizes the context from p-blocks to C -blocks, but which, when specialized again to p-blocks is a bit weaker than the definition of M. Broué (cf [2]).

Broué’s definition goes as follows: we let G and H be two finite groups, (K, O, k) be a p-modular system large enough for both G and H, e and f be central idempotents of OG and OH respectively, and we set A = OGe and B = OHf. To any generalized character μ of G × H (written μ ∈ ZIrr(K(G×H))), we can associate a linear map I_{μ}: ZIrr(KH) ⟶ ZIrr(KG) by letting, for ζ ∈ Irr(KH) and g ∈ G,

I_{μ}(ζ)(g) = \frac{1}{|H|} ∑_{h∈H} μ(g, h^{−1})ζ(h).

Definition 2.1. (cf [2]) A generalized character μ of G × H, is perfect if the following two conditions are satisfied: (per. 1) For all g ∈ G and h ∈ H, |C_{G}(g)|_{p} and |C_{H}(h)|_{p} divide μ(g, h). (per. 2) If μ(g, h) ≠ 0, then g and h are either both p-regular, or both psingular. If furthermore the map I_{μ} defined by μ induces a bijective isometry between ZIrr(KB) and ZIrr(KA), then I_{μ} is said to be a perfect isometry between B and A, and B and A are said to be perfectly isometric.

We then have the following (cf [2]):

Theorem 2.2. Suppose that, with the notations above, B and A are perfectly isometric via I_{μ}. Then:

(i) I_{μ} defines a bijection between primitive idempotents of the centers Z(KHf) and Z(KGe), which in turn induces an algebra isomorphism between Z(OHf) and Z(OGe). (ii) I_{μ} induces a bijection between the blocks of H and G associated to f and e respectively, which preserves the defect and number of ordinary irreducible characters, the number of irreducible Brauer characters, the height of ordinary irreducible characters, and the elementary divisors of the Cartan matrix.

In particular, under the same hypotheses, there exist a bijection I: Irr(B) ⟶ Irr(A), and signs {ε(ζ), ζ ∈ Irr(B)} such that, for all ζ ∈ Irr(B), I_{μ}(ζ) = ε(ζ)I(ζ).

We now turn to the definition introduced in [18]. If G and H are finite groups, C and D are closed unions of conjugacy classes of G and H respectively, and if b (resp. b′) is a union of C-blocks of G (resp. D-blocks of H), then we say that there is a generalized perfect isometry between b and b′ (with respect to C and D) if there exists a bijection with signs between b and b′, which furthermore preserves contributions; i.e. there exists a bijection I: b ⟼ b′ such that, for each χ ∈ b, there is a sign ε(χ), and such that

∀χ, ψ ∈ b, < I(χ), I(ψ) >_{D}=< ε(χ)χ, ε(ψ)ψ >_{C} .

Note that this is equivalent to < I(χ), I(ψ) >_{D′}=< ε(χ)χ, ε(ψ)ψ >_{C′}, where C′ = G \ C and D′ = H \ D.

We define R(C, b) to be the Z-submodule of the space of complex class functions of G generated by {χ^{C} | χ ∈ b}, and P(C, b) to be the Z-submodule of R(C, b) consisting of generalized characters. The fact that C is closed implies (using Galois theory) that the modules R(C, b) and P(C, b) have the same Z-rank, and that this rank is the number of conjugacy classes in C(cf [24]). Given a Z-basis {φ_{1}, …, φ_{s}} for P(C, b), we let C(b) be the s × s matrix with (i, j)-entry < φ_{i}, φ_{j} >_{C}, and call C(b) the Cartan matrix of b. A different choice of Z-basis for P(C, b) leads to a Cartan matrix with the same elementary divisors. We define similarly R(D, b′), P(D, b′) and C(b′). We then have the following:

Theorem 2.3. (Proposition 1.4 in [18]) With the above notations, if there is a generalized perfect isometry between b and b′ with respect to C and D, then the Abelian groups R(C, b) and R(D, b′) are isomorphic via an isomorphism which restricts to an isomorphism between P(C, b) and P(D, b′). With suitable choice of Z-bases, the Cartan matrices C(b) and C(b′) are equal.

In particular, we see that, if C (resp. D) is the set of p-regular elements of G (resp. H), then such a generalized perfect isometry induces a bijection between the p-blocks in b and b′, which preserves the numbers of ordinary irreducible characters and irreducible Brauer characters, and the elementary divisors of the Cartan matrix. However, with this more general definition, one doesn’t get the algebra isomorphism of Theorem 2.2 (i).

One of Broué’s conjectures (which is just the shadow, at the level of characters, of much deeper equivalences conjectured by Broué) states that, if G is a finite group with Abelian Sylow p-subgroup U, and if B = N_{G}(U) is the normalizer of U in G, then the principal p-blocks of B and G are perfectly isometric (in Broué’s sense).

This conjecture doesn’t apply to the three families of groups of Lie rank one we will consider when p is the defining characteristic, for then the Sylow p-subgroups are not Abelian. Furthermore, it is shown in Cliff [8] that, if G is a Suzuki group, then there isn’t any perfect isometry between the principal 2-blocks of B and G. This was the starting point of this work. In this part, we show that, in the case of Suzuki groups and in two other (and quite similar) families, there is a generalized perfect isometry between the principal p-blocks of B and G, with respect to p^{2}-regular (or p^{2}-singular) elements.

Section 2.3 deals with special unitary groups SU(3, q^{2}), section 2.4 with Suzuki groups Sz(q), and section 2.5 with Ree groups (of type G_{2}) Re(q).

In two of the families of groups we are studying, we will use two theorems concerning p-blocks. The first, due to Green, states that any defect group of a p-block of a (finite) group G is the intersection of two Sylow p-subgroups of G (see e.g. [20], Corollary (4.21)). An immediate consequence of this is that, if the Sylow p-subgroups of G have trivial intersection, then any pblock of G has either maximal defect or defect 0. The second result, proved by Fong in the case of p-solvable groups (see e.g. Isaacs [16], Problem (4.9)), but which still holds in general, states that, if a finite group G is such that C_{G}(O_{p}(G)) ≤ O_{p}(G), where O_{p}(G) is the largest normal p-subgroup of G, then G has exactly one p-block.

2.3 Special Unitary Groups

In this section, we denote by G = SU(3, q^{2}) the 3-dimensional special unitary group on F_{q^{2}}, where q = p^{n} for some prime p and n ≥ 1. We write d = gcd(3, q + 1). Then the center Z(G) of G is cyclic of order d. Note that, if p = 3, then d = 1. Thus, whatever the value of p, the integers p and d are coprime, and their least common multiple is pd. The order of G is |G| = q^{3}(q + 1)^{2}(q − 1)(q^{2} − q + 1). We consider U a Sylow p-subgroup of G and its normalizer B = N_{G}(U), which is a semi-direct product of U with a cyclic group H of order q^{2} − 1. We have |U| = q^{3} and |B| = q^{3}(q^{2} − 1). For a complete description of G and B, their conjugacy classes and ordinary and modular characters, we refer to Geck [13], whose notations are now in force.

2.3.1 Conjugacy classes

The tables of conjugacy classes of G and B are taken from Geck [13]. Only the order of their elements have been added (these can be easily computed using the canonical forms given in [13]).

The group G has q^{2}+q+1+d^{2} conjugacy classes, parametrized as follows:

name  parameters  length  order
C_{1}^{(k)}  0 ≤ k ≤ d − 1  1  1 (k = 0) or d
C_{2}^{(k)}  0 ≤ k ≤ d − 1  (q^{3} + 1)(q − 1)  p (k = 0) or pd
C_{3}^{(k,l)}  0 ≤ k, l ≤ d − 1  \frac{1}{d}q(q^{2} − 1)(q^{3} + 1) 2p (k = 0) or 2dp
C_{4}^{(k)}  \begin{matrix}0 ≤ k ≤ q \\ \frac{q+1}{d} ∤ k \end{matrix}  q^{2}(q^{2} − q + 1)  \frac{q+1}{gcd(k, q+1)}
C_{5}^{(k)}  \begin{matrix}0 ≤ k ≤ q \\ \frac{q+1}{d} ∤ k \end{matrix}  q^{2}(q − 1)(q^{3} + 1)  \frac{2(q+1)}{gcd(k, q+1)}
C_{6}^{(k,l,m)} \begin{matrix}0 ≤ k < l < m ≤ q \\ k + l + m ≡ 0(mod q + 1) \end{matrix} q^{3}(q − 1)(q^{2} − q + 1) dividing q + 1

C_{7}^{(k)}

0 ≤ k ≤ q^{2} − 2
k ≢ 0(mod q − 1);
if k_{1} ≡ −qk(mod q^{2} − 1)
then C_{7}^{(k)} = C_{7}^{(k_{1})}

q^{3}(q^{3} + 1) \frac{(q+1)(q−1)}{gcd((q+1)(q−1), k)}

C_{8}^{(k)}

0 ≤ k ≤ q^{2} − q
\frac{q^{2}−q+1}{d} ∤ k;
if k_{1} ≡ −kq or k_{2} ≡ kq^{2}
(mod q^{2} − q + 1), then
C_{8}^{(k)} = C_{8}^{(k_{1})} = C_{8}^{(k_{2})}

q^{3}(q + 1)^{2}(q − 1) \frac{q^{2}−q+1}{gcd(q^{2}−q+1, k)}

B has q^{2} + q + d conjugacy classes, parametrized as follows:

name fusion into G  parameters  length
B_{1}^{(k)}  C_{1}^{(k)}  0 ≤ k ≤ d − 1  1
B_{2}^{(k)}  C_{2}^{(k)}  0 ≤ k ≤ d − 1  q − 1
B_{3}^{(k,l)}  C_{3}^{(k,l)}  0 ≤ k, l ≤ d − 1  \frac{1}{d}q(q^{2} − 1)
B_{4}^{(k)}  C_{4}^{(k)}  0 ≤ k ≤ q , \frac{q+1}{d} |̸k  q^{2}
B_{5}^{(k)}  C_{5}^{(k)}  0 ≤ k ≤ q , \frac{q+1}{d} |̸k  q^{2}(q − 1)
B_{6}^{(k)}  C_{7}^{(k)}  0 ≤ k ≤ q^{2} − 2 , q − 1 |̸k  q^{3}

We are only interested in elements of order divisible by p^{2}. We see from the tables above that, unless p = 2, there is no such element in G.

Remark: this can be seen without actually computing the orders of the elements of G. Take any p-element g in G, of order p^{k} say. Then its eigenvalues, which lie in some finite extension of F_{q^{2}}, F_{p^{l}} say, must be p^{k}-th roots of 1.

But the only p^{k}-th root of 1 in F_{p^{l}} (in which each x is a root of x^{p^{l}} − x) is 1 itself. Thus the characteristic polynomial of g is (X − 1)^{3}, and thus (g − Id)^{3} = 0. Then, if p ≥ 3, then g^{p} − Id = (g − Id)^{p} = 0, so that g^{p} = Id. Hence, if p ≥ 3, then any non-trivial p-element of G has order p.

As a consequence, if p ≠ 2, then every element of G is p^{2}-regular, so that the scalar product on p^{2}-regular elements is just the ordinary scalar product, and there exists a generalized perfect isometry between the principal p-blocks B_{0} and b_{0} of G and B with respect to p^{2}-regular elements if and only if there exists a bijection between B_{0} and b_{0} (any bijection will give a generalized perfect isometry).

If p = 2, we will only consider the values of irreducible characters on elements of order divisible by 4, that is, we will consider

C = {g ∈ G | 4|o(g)} = ⋃_{0≤k, l≤d−1} C_{3}^{(k,l)}

and

D = {h ∈ B | 4|o(h)} = ⋃_{0≤k, l≤d−1} B_{3}^{(k,l)}.

Both C and D are closed unions of conjugacy classes (of G and B respectively).

2.3.2 Irreducible characters, principal blocks

The irreducible characters of G can be listed as follows (The index used in the notation for a character indicates its degree; characters of the same degree are then parametrized by parameters u and v. Note that for characters of degree q^{3} + 1 and (q + 1)^{2}(q − 1), different choices of parameters can yield the same character (cf [13]); we therefore indicate the number of characters in these families.) :

χ_{1}, χ_{q^{2}−q}, χ_{q^{3}}, {χ_{q^{2}−q+1}^{(u)}, χ_{q(q^{2}−q+1)}^{(u)} | 1 ≤ u ≤ q},
{χ_{(q−1)(q^{2}−q+1)}^{(u,v)} | 1 ≤ u ≤ (q + 1)/3, u < v < 2(q + 1)/3},
{χ_{q^{3}+1}^{(u)} | 1 ≤ u ≤ q^{2} − 1, q − 1 ∤ u} ( (q + 1)(q − 2)/2 characters),

{χ_{(q+1)^{2}(q−1)}^{(u)} | 0 ≤ u ≤ q^{2} − q, q^{2} − q + 1 ∤ u} ( (q^{2} − q + 1 − d)/3 characters),

and, if d = 3, we have in addition

{χ_{(q−1)(q^{2}−q+1)/3}^{(u)} | 0 ≤ u ≤ 2} and {χ_{(q+1)^{2}(q−1)/3}^{(u,v)} | 0 ≤ u ≤ 2, v = 1, 2}.

The irreducible characters of B can be listed as follows:

{θ_{1}^{(u)} | 0 ≤ u ≤ q^{2} − 2}, {θ_{q(q−1)}^{(u)} | 0 ≤ u ≤ q}, {θ_{q^{2}−1}} (only if d = 1),

and {θ_{(q^{2}−1)/3}^{(u,v)} | 0 ≤ u, v ≤ 2} (only if d = 3).

We now compute the principal p-blocks B_{0} and b_{0} of G and B respectively. We compute them directly, using the fact that two (complex) irreducible

characters belong to the same p-block if and only if the associated central characters are the same when reduced to a field of characteristic p (cf [20], definition (3.1) and Problem (3.3)):

Theorem 2.4. If N is a finite group, (K, O, k) a p-modular system large enough for H, O being the localization of A_{K} (algebraic integers of K) in P, then χ ∈ Irr(KN) belongs to the principal p-block of N if and only if

|C|\frac{χ(C)}{χ(1)} ≡ |C| (mod P)

for each p-regular conjugacy class C of N.

Most of the work is done using the following observation: with the notations of the previous theorem, and given C a conjugacy class of N, if |C| is divisible by p, then |C| ∈ P (since P ∩ Z = pZ), so that we only need to check whether |C|\frac{χ(C)}{χ(1)} ∈ P. We distinguish two special cases:

a) If χ(C) = 0, then the property is clearly verified.

b) If \frac{|C|}{χ(1)} ∈ P, then, since χ(C) ∈ A_{K} and P ⊲ A_{K}, |C|\frac{χ(C)}{χ(1)} ∈ P.

It turns out that, for any p-regular conjugacy class of length divisible by p and any irreducible character of G or B, one of these two cases apply:

We first consider the group G. The p-regular conjugacy classes of length divisible by p are those of type C_{4}, C_{6}, C_{7} and C_{8}. We examine only characters distinct from χ_{1} and χ_{q^{3}} (the first one belonging to B_{0} and the second one being of defect 0 and thus not belonging to B_{0}). C_{8} has length q^{3}(q +1)^{2}(q −1), so that case b) applies to characters of degree q^{2} − q, (q + 1)^{2}(q − 1) and (q + 1)^{2}(q − 1)/3; case a) applies to any other character. C_{7} has length q^{3}(q^{3} + 1) = q^{3}(q + 1)(q^{2} − q + 1), so that case b) applies to characters of degree q^{2} − q + 1, q(q^{2} − q + 1) and q^{3} + 1; case a) applies to any other character. C_{6} has length q^{3}(q − 1)(q^{2} − q + 1), so that case b) applies to characters of degree q^{2}−q, q^{2}−q+1, q(q^{2}−q+1), (q−1)(q^{2}−q+1) and (q−1)(q^{2}−q+1)/3; case a) applies to any other character.

C_{4} has length q^{2}(q^{2} − q + 1), so that case b) applies to characters of degree q^{2}−q+1 and q(q^{2}−q+1); case a) applies to characters of degree (q+1)^{2}(q−1) and (q + 1)^{2}(q − 1)/3; we check the other cases by hand :

|C_{4}|\frac{χ_{q^{2}−q}(C_{4})}{χ_{q^{2}−q}(1)} = \frac{q^{2}(q^{2} − q + 1)(1 − q)}{q^{2} − q} = −q(q^{2} − q + 1) ∈ P

|C_{4}|\frac{χ_{(q−1)(q^{2}−q+1)}(C_{4})}{χ_{(q−1)(q^{2}−q+1)}(1)} = \frac{q^{2}(q^{2} − q + 1)(q − 1)α}{(q − 1)(q^{2} − q + 1)} = q^{2}α ∈ P (α ∈ A_{K})

|C_{4}|\frac{χ_{q^{3}+1}(C_{4})}{χ_{q^{3}+1}(1)} = \frac{q^{2}(q^{2} − q + 1)(q + 1)α′}{(q^{3} + 1)} = q^{2}α′ ∈ P (α′ ∈ A_{K})

|C_{4}|\frac{χ_{(q−1)(q^{2}−q+1)/3}(C_{4})}{χ_{(q−1)(q^{2}−q+1)/3}(1)} = \frac{q^{2}(q^{2} − q + 1)(q − 1)}{(q − 1)(q^{2} − q + 1)/3} = 3q^{2} ∈ P.

Now, in B, the p-regular conjugacy classes of length divisible by p are those of type B_{4} and B_{6}. We have

\frac{|B_{4}|}{θ_{1}(1)} = q^{2} ∈ P and \frac{|B_{6}|}{θ_{1}(1)} = q^{3} ∈ P

so that case b) applie to these two cases. Furthermore,

|B_{4}|\frac{θ_{q(q−1)}(B_{4})}{θ_{q(q−1)}(1)} = \frac{q^{2}(1 − q)α′′}{q(q − 1)} = −qα′′ ∈ P (α′′ ∈ A_{K})

and, in every other cases, case a) applies.

In order to compute the principal blocks B_{0} and b_{0}, we thus only have to study the values of irreducible characters on p-regular conjugacy classes of G and B of length not divisible by p. These are the C_{1}^{(k)}’s and the B_{1}^{(k)}’s respectively, and they have length 1 (since lie in the center). For C one of these classes, and ψ an irreducible character of the corresponding group, we thus have to check whether \frac{ψ(C)}{ψ(1)} is congruent to 1 modulo P. But, since C

lies in the center, which has order d, \frac{ψ(C)}{ψ(1)} is a d-th root of 1. And it can be shown that, if ε is a p′-root of 1, then Z[ε] ∩ P = pZ[ε]. In particular, we get that \frac{ψ(C)}{ψ(1)} − 1 ∈ P if and only if \frac{ψ(C)}{ψ(1)} = 1.

Principal block of B

We see that, for fixed degree, the value of θ ∈ Irr(B) on B_{1}^{(k)} only depends on the parameter u in θ (and on k); writing β = e^{2iπ/d}, we get, in all the cases,

θ_{u} ∈ b_{0} ⟺ (β^{uk} = 1 for all 0 ≤ k ≤ d − 1).

This is always true if d = 1 (for then β = 1), so that b_{0} = Irr(B) and |b_{0}| = q^{2} + q + 1. If d = 3, then β = e^{2iπ/3}, and thus

θ_{u} ∈ b_{0} ⟺ β^{0} = β^{u} = β^{2u} = 1
⟺ β^{u} = 1
⟺ u ≡ 0 (mod 3)

Hence

b_{0} = {θ_{1}^{(u)} | 0 ≤ u ≤ q^{2} − 2, 3|u} ∪ {θ_{q(q−1)}^{(u)} | 0 ≤ u ≤ q, 3|u}
∪{θ_{(q^{2}−1)/3}^{(u,v)} | 0 ≤ u, v ≤ 2, 3|u}
= {θ_{1}^{(u)} | 0 ≤ u ≤ q^{2} − 2, 3|u} ∪ {θ_{q(q−1)}^{(u)} | 0 ≤ u ≤ q, 3|u}
∪{θ_{(q^{2}−1)/3}^{(0,v)} | 0 ≤ v ≤ 2}

Now d = gcd(3, q + 1) = 3, so that q ≡ −1(mod 3), q^{2} ≡ 1(mod 3) and q^{2} − 2 ≡ −1(mod 3), and thus |b_{0}| = \frac{q^{2}−1}{3} + \frac{q+1}{3} + 3 = \frac{q(q+1)}{3} + 3.

Principal block of G Similarly, we must check, for χ ∈ Irr(G) \ {χ_{q^{3}}} and 0 ≤ k ≤ d − 1,

whether \frac{χ(C_{1}^{(k)})}{χ(1)} = 1. We let again β = e^{2iπ/d}.

If d = 1, then β = 1, and this is true for all χ ∈ Irr(G) \ {χ_{q^{3}}}, so that B_{0} = Irr(G) \ {χ_{q^{3}}} and |B_{0}| = q^{2} + q + 2 − 1 = q^{2} + q + 1. If d = 3, then β = e^{2iπ/3}. The result is true for χ_{1}, χ_{q^{2}−q} and χ ∈ {χ_{(q−1)(q^{2}−q+1)/3}^{(u)} | 0 ≤ u ≤ 2}, whose values on the C_{1}^{(k)}’s do not depend on d.

For χ_{u} ∈ Irr(G) \ ({χ_{1}, χ_{q^{3}}, χ_{q^{2}−q}} ∪ {χ_{(q−1)(q^{2}−q+1)}^{(u,v)}} ∪ {χ_{(q+1)^{2}(q−1)/3}^{(u,v)}}

∪{χ_{(q−1)(q^{2}−q+1)/3}^{(u)}}), we have

\begin{aligned}χ_{u} ∈ B_{0} & ⟺ β^{uk} = 1 for all k = 0, 1, 2 \\ & ⟺ 3|u. \end{aligned}

Finally

\begin{aligned}χ_{(q−1)(q^{2}−q+1)}^{(u,v)} ∈ B_{0} & ⟺ β^{(u+v)k} = 1 for all k = 0, 1, 2 \\ & ⟺ 3|u + v \end{aligned}

and

\begin{aligned}χ_{(q+1)^{2}(q−1)/3}^{(u,v)} ∈ B_{0} & ⟺ β^{vk} = 1 for all k = 0, 1, 2 \\ & ⟺ 3|v, \end{aligned}

which is excluded.

Hence

B_{0} = {χ_{1}, χ_{q^{2}−q}} ∪ {χ_{q^{2}−q+1}^{(u)}, χ_{q(q^{2}−q+1)}^{(u)} | 1 ≤ u ≤ q, 3|u} ∪ {χ_{(q−1)(q^{2}−q+1)}^{(u,v)} |

1 ≤ u ≤ (q+1)/3, u < v < 2(q+1)/3, 3|u+v}∪{χ_{q^{3}+1}^{(u)} | 1 ≤ u ≤ q^{2}−1 (…), 3|u}

∪{χ_{(q+1)^{2}(q−1)}^{(u)} | 0 ≤ u ≤ q^{2} − q (…), 3|u} ∪ {χ_{(q−1)(q^{2}−q+1)/3}^{(u)} | 0 ≤ u ≤ 2}

and it can be verified that, in this case, |B_{0}| = 2 + 2(\frac{q+1}{3} − 1) + \frac{(q+1)(q−2)}{18} + \frac{(q+1)(q−2)}{6} + \frac{q^{2}−q−2}{9} + 3 = q \frac{q+1}{3} + 3.

We thus notice that |b_{0}| = |B_{0}|, independantly of d. In particular, if p ≠ 2, then there is a generalized perfect isometry between B_{0} and b_{0} with respect to p^{2}-regular elements. In the rest of this section, we will therefore only consider the case p = 2.

2.3.3 A generalized perfect isometry

We start by observing that the number of conjugacy classes of elements of order divisible by 4 is the same in G and B. Moreover, if x ∈ B is such an element, then |C_{G}(x)| = |C_{B}(x)| = dq^{2}. As a consequence, if we can find a bijection with signs between B_{0} and b_{0} which furthermore preserves the values of characters on 4-singular elements, then it will also preserve contributions, and thus will be a generalized perfect isometry between B_{0} and b_{0} with respect to 4-singular elements (or 4-regular elements).

If d = 1, we have the following fragment of the character table of B:

θ_{q(q−1)}^{(u)}, 0 ≤ u ≤ q θ_{1}^{(u)}, 0 ≤ u ≤ q^{2} − 2 θ_{q^{2}−1}
B_{3}^{(0,0)}  0  1  −1

and the corresponding fragment for G:

χ_{q^{2}−q} \begin{matrix}χ_{q(q^{2}−q+1)}^{(u)} \\ 1 ≤ u ≤ q \end{matrix} \begin{matrix}χ_{q^{2}−q+1}^{(u)} \\ 1 ≤ u ≤ q \end{matrix} \begin{matrix}χ_{(q−1)(q^{2}−q+1)}^{(u,v)} \\ 1 ≤ u ≤ (q + 1)/3 \\ u < v < 2(q + 1)/3 \end{matrix}
C_{3}^{(0,0)} 0  0  1  −1

\begin{matrix}χ_{q^{3}+1}^{(u)} \\ 1 ≤ u ≤ q^{2} − 1 \\ q − 1 ∤ u \end{matrix} \begin{matrix}χ_{(q+1)^{2}(q−1)}^{(u)} \\ 0 ≤ u ≤ q^{2} − q \\ q^{2} − q + 1 ∤ u \end{matrix} χ_{1}
C_{3}^{(0,0)}  1  −1  1

In view of the remark above, we can see that there is a generalized perfect isometry between b_{0} and B_{0}. Indeed, listing the irreducible characters of b_{0} and B_{0} in the same order as in the above tables, and relabelling them so that Irr(b_{0}) = {Θ_{1}, …, Θ_{q^{2}+q+1}} and Irr(B_{0}) = {X_{1}, …, X_{q^{2}+q+1}}, then a generalized perfect isometry between b_{0} and B_{0} is given by

I: Θ_{i} ⟼ ε_{i}X_{i} , i = 1, … q^{2} + q + 1,

where ε_{i} = −1 if X_{i} has degree (q − 1)(q^{2} − q + 1), (q + 1)^{2}(q − 1) or 1, and ε_{i} = 1 otherwise.

If d = 3, we have the following fragment of the character table of B:

\begin{matrix}θ_{q(q−1)}^{(u)} \\ 0 ≤ u ≤ q, 3|u \end{matrix} \begin{matrix}θ_{1}^{(u)} \\ 0 ≤ u ≤ q^{2} − 2, 3|u \end{matrix} \begin{matrix}θ_{(q^{2}−1)/3}^{(0,v)} \\ 0 ≤ v ≤ 2 \end{matrix}
\begin{matrix}B_{3}^{(k,l)} \\ 0 ≤ k, l ≤ 2 \end{matrix}  0  1  qδ_{vl} − \frac{q+1}{3}

and the corresponding fragment for G:

χ_{q^{2}−q} \begin{matrix}χ_{q(q^{2}−q+1)}^{(u)} \\ 1 ≤ u ≤ q \\ 3|u \end{matrix} χ_{1} \begin{matrix}χ_{q^{2}−q+1}^{(u)} \\ 1 ≤ u ≤ q \\ 3|u \end{matrix} \begin{matrix}χ_{(q−1)(q^{2}−q+1)}^{(u,v)} \\ 1 ≤ u ≤ (q + 1)/3 \\ u < v < 2(q + 1)/3 \\ 3|(u + v) \end{matrix}

\begin{matrix}C_{3}^{(k,l)} \\ 0 ≤ k, l ≤ 2 \end{matrix}  0  0  1  1  −1

\begin{matrix}χ_{q^{3}+1}^{(u)} \\ 1 ≤ u ≤ q^{2} − 1 \\ q − 1 ∤ u \\ 3|u \end{matrix} \begin{matrix}χ_{(q+1)^{2}(q−1)}^{(u)} \\ 0 ≤ u ≤ q^{2} − q \\ q^{2} − q + 1 ∤ u \\ 3|u \end{matrix} \begin{matrix}χ_{(q−1)(q^{2}−q+1)/3}^{(u)} \\ 0 ≤ u ≤ 2 \end{matrix}

\begin{matrix}C_{3}^{(k,l)} \\ 0 ≤ k, l ≤ 2 \end{matrix}  1  −1  qδ_{lu} − \frac{q+1}{3}

Then, here again, we can see that b_{0} and B_{0} are perfectly isometric (with respect to 4-regular elements): listing the irreducible characters of b_{0} and B_{0} in the same order as in the above tables, and relabelling them so that Irr(b_{0}) = {Θ_{1}, …, Θ_{q(q+1)/3+3}} and Irr(B_{0}) = {X_{1}, …, X_{q(q+1)/3+3}}, then we have the following generalized perfect isometry between b_{0} and B_{0}:

I: Θ_{i} ⟼ ε_{i}X_{i} , i = 1, … q(q + 1)/3 + 3,

where ε_{i} = −1 if X_{i} has degree (q − 1)(q^{2} − q + 1) or (q + 1)^{2}(q − 1), and ε_{i} = 1 otherwise.

2.4 Suzuki Groups

In this section, we let G = Sz(q) be a Suzuki group, where q = 2^{2n+1} for some n ≥ 1, U a Sylow 2-subgroup of G, and B = N_{G}(U).

2.4.1 Conjugacy classes

The conjugacy classes of G can be found in Burkhardt [4]. We have |G| = q^{2}(q − 1)(q^{2} + 1), and, writing r^{2} = 2q (r ∈ N), we have representatives for the conjugacy classes of G : t (order 2), f, f^{−1} (order 4) for the elements of even order, and

1, \{x^{a}, 1 ≤ a ≤ \frac{q − 2}{2}\}, \{y^{b}, 1 ≤ b ≤ \frac{q + r}{4}\}, \{z^{c}, 1 ≤ c ≤ \frac{q − r}{4}\}

for the elements of odd order (dividing q−1, q+r+1 and q−r+1 respectively). We have |C_{G}(f)| = |C_{G}(f^{−1})| = 2q. For the following facts about the structure of B and references for proofs,

we refer to Cliff [8].

We have |U| = q^{2}, and B = N_{G}(U) is a semi-direct product of U by a cyclic group A of order q − 1. U is a TI (trivial intersection) set in G with respect to B, and B is a Frobenius group of order q^{2}(q−1) (U is the Frobenius kernel and A is the Frobenius complement). Letting C be the center of U, we have C ≅ U/C ≅ (F_{q}, +), and A acts regularly on the sets of non-identity elements of both C and U/C. We have the following non-trivial conjugacy classes in B :

- a unique class of involutions, consisting of the non-identity elements of C. For τ in this class, |C_{B}(τ)| = q^{2}.

- two classes of elements of order 4, Cl_{B}(ρ) and Cl_{B}(ρ^{−1}), where ρ ∈ U \C. We have |C_{B}(ρ)| = |C_{B}(ρ^{−1})| = 2q.

- q − 2 classes of elements of odd order : {Cl_{B}(π), π ∈ A \ {1}}. For such a π, C_{B}(π) = A has order q − 1.

We can now compute the conjugacy classes of U. For any involution τ in U, we have τ ∈ C = Z(U) and Cl_{U}(τ) = {τ}, which gives us q − 1 classes of involutions: {{τ_{j}}, 2 ≤ j ≤ q}. Using the fact that the centralizer in B of an element of U of order 4 is a 2-group, and thus is equal to its centralizer in U, we see that U must have 2(q − 1) classes of elements of order 4, each of length q/2 : {Cl_{U}(f_{2}), Cl_{U}(f_{2}^{−1}), …, Cl_{U}(f_{q}), Cl_{U}(f_{q}^{−1})}, where (without loss of generality) f_{j}^{2} = τ_{j} for all j = 2, …, q. This makes a total of 3q − 2 conjugacy classes in U. Furthermore, C = Z(U) is also the commutator subgroup of U, so that U has [U : U′] = [U : C] = q linear irreducible (complex) characters.

2.4.2 Irreducible characters of B

First step and Clifford’s Theorem

From the semi-direct product structure, we have a group homomorphism π: B ⟶ A, which gives us q − 1 distinct linear (irreducible) characters of B with U in their kernel : π ∘ λ_{1} = 1_{B}, π ∘ λ_{2} = φ_{2}, …, π ∘ λ_{q−1} = φ_{q−1}, where Irr(A)= {λ_{1} = 1_{A}, λ_{2}, …, λ_{q−1}}. Therefore 3 elements μ_{1}, μ_{2}, μ_{3} of Irr(B) are missing, and these don’t have U in their kernel (for otherwise they could be lifted to B from irreducible characters of A). Whence, since B is a Frobenius group with Frobenius kernel U, μ_{1}, μ_{2} and μ_{3} are induced by non-trivial irreducible characters of U (cf Curtis-Reiner [9], Proposition (14.4)). We write μ_{i} = Ind_{U}^{B}(ν_{i}) for some ν_{i} ∈ Irr(U) \ {1_{U}}, i = 1, 2, 3. Writing |B| = q^{2}(q − 1) = (q − 1)1 + μ_{1}(1)^{2} + μ_{2}(1)^{2} + μ_{3}(1)^{2}, we see that ν_{1}(1)^{2}+ν_{2}(1)^{2}+ν_{3}(1)^{2} = q+1 odd. The ν_{i}(1)’s being powers of 2, this implies that, without loss of generality, ν_{1}(1) = 1, μ_{1}(1) = q−1 and ν_{2}(1)^{2}+ν_{3}(1)^{2} = q = 2^{2n+1}, so that ν_{2}(1) = ν_{3}(1) = 2^{n} and μ_{2}(1) = μ_{3}(1) = (q − 1)2^{n}.

Finally, B has the following irreducible characters:

\begin{matrix}φ_{1} = 1_{B}, …, φ_{q−1} \\ degree 1 \end{matrix}  \begin{matrix}μ_{1} \\ degree q − 1 \end{matrix}  \begin{matrix}μ_{2}, μ_{3} \\ degree (q − 1)2^{n} \end{matrix}

\begin{matrix}φ_{i} = π ∘ λ_{i}, \\ i = 1, … q − 1, \\ π: B = U ⋊ A ↠ A, \\ Irr(A) = \{λ_{1}, …, λ_{q−1}\}. \end{matrix} \begin{matrix}μ_{1} = Ind_{U}^{B}(ν_{1}), \\ ν_{1} ∈ Irr(U) \ \{1_{U}\}, \\ ν_{1}(1) = 1. \end{matrix} \begin{matrix}μ_{i} = Ind_{U}^{B}(ν_{i}), \\ ν_{i} ∈ Irr(U) \ \{1_{U}\}, \\ ν_{i}(1) = 2^{n}, \\ i = 2, 3. \end{matrix}

Now, U acts (trivially) by conjugation on C, and thus on Irr(C). By Clifford’s Theorem, we see that, if χ ∈ Irr(U) and if < Res_{C}^{U}(χ), μ >_{C}≠ 0 for some μ ∈ Irr(C), then Res_{C}^{U}(χ) = eμ for some e ∈ N. Furthermore, since every element of Irr(C) is linear, e = χ(1), a power of 2. In particular, Res_{C}^{U}(ν_{1}) = α_{1}, Res_{C}^{U}(ν_{2}) = 2^{n}α_{2} and Res_{C}^{U}(ν_{3}) = 2^{n}α_{3} for some α_{1}, α_{2}, α_{3} linear (irreducible) characters of C.

Furthermore, using Frobenius Reciprocity, we see that we can define an injective map

Irr(C) ⟶ Irr(U)
α ⟼ ν s.t. < Ind_{C}^{U}(α), ν >_{U}≠ 0

Actions of A and B

We have C ⊲ B, and B acts by conjugation on the conjugacy classes of C: for c ∈ C, u ∈ U and a ∈ A, we have (u, a)^{−1}(c, 1)(u, a) = (c^{a^{−1}}, 1) where c^{a^{−1}} denotes the image of c under a^{−1} in the action of A on U (which action preserves C which is characteristic in U). Hence the action of B in fact reduces to an action of A (by this we mean that {B-orbits of C} = {A-orbits of C}).

Now B also acts by conjugation on Irr(C) = {β_{1} = 1_{C}, β_{2}, …, β_{q}} via

β_{i}^{h}(c) = β_{i}(c^{h^{−1}}) forc ∈ C, h ∈ B, and i = 1, …, q,

where c^{h^{−1}} denotes the image of c under h^{−1} in the B-action we just described. And, here again, {B-orbits of Irr(C)} = {A-orbits of Irr(C)}. We can now apply Corollary (11.10) in Curtis-Reiner [9] which states that, if B acts by conjugation on the conjugacy classes of C⊲B and on Irr(C), then the number of B-orbits of these two actions coincide. But, in the first action, we only have two orbits : {1} and C \ {1} (since A acts regularly on C \ {1}). Since {1_{C}} is clearly an orbit in the action on Irr(C), we must have

{β_{2}^{a}, a ∈ A} = {β_{2}, … β_{q}}(= {β_{j}^{a}, a ∈ A} for all j = 2, … q).

And, since |A| = q − 1, for all i = 2, … q, if β_{i}^{a} = β_{i}^{b}, then a = b.

Now B acts by conjugation on Irr(U) and, the (underlying) action of U being trivial, this also reduces to an action of A. Furthermore, for any χ, ψ ∈ Irr(U), for all a ∈ A, since < χ^{a}, ψ^{a} >_{U}=< χ, ψ >_{U}, we have

χ^{a} = ψ^{a} ⟺ χ = ψ (△)

Now take any i ∈ {2, … q}. Then Ind_{C}^{U}(β_{i}) has, say, k distinct irreducible summands in Irr(U). If χ ∈ Irr(U) is any of these, then, for all a ∈ A, χ^{a} is an irreducible summand of Ind_{C}^{U}(β_{i}^{a}). This also implies that, for a, b ∈ A, χ^{a} = χ^{b} ⇒ β_{i}^{a} = β_{i}^{b} ⇒ a = b. Hence the orbit of χ under the action of A has size q − 1 (and all the elements in this orbit have the same degree).

Now, because of (△), we see that, for all a ∈ A, Ind_{C}^{U}(β_{i}^{a}) must have at least k distinct irreducible summands in Irr(U). And, since i was arbitrary, and since {β_{i}^{a}, a ∈ A} = {β_{2}, … β_{q}}, we get that, for any i ∈ {2, … q}, Ind_{C}^{U}(β_{i}) has exactly k irreducible summands in Irr(U). Notice that all the elements of Irr(U) which arose are distinct because of the injectivity of the map we constructed in the last paragraph. Since all irreducible characters of U appear as irreducible summands of characters induced from C to U, this gives us

|Irr(U)| = |{irr. summ. of Ind_{C}^{U}(1_{C})}| + (q − 1)k.

Irreducible characters of U

By Frobenius Reciprocity, 1_{U} is an irreducible summand of Ind_{C}^{U}(1_{C}). Furthermore, by Frobenius Reciprocity and Clifford’s Theorem, we can write Ind_{C}^{U}(1_{C}) = ∑ χ(1)χ. In particular, any linear irreducible summand of Ind_{C}^{U}(1_{C}) appears with multiplicity 1.

Thus ∑(χ(1))^{2} = q even yields that there must be (at least) one other (than 1_{U}) linear irreducible summand of Ind_{C}^{U}(1_{C}), α say.

Now U has exactly q linear irreducible characters. Supposing one of these is an irreducible summand of some Ind_{C}^{U}(β_{i}), β_{i} ≠ 1_{C}, then we would find one distinct linear irreducible character in each of {Ind_{C}^{U}(β_{j}) | j ≠ i, β_{j} ≠ 1_{C}}, each distinct from 1_{U} and α, thus giving 2 + q − 1 = q + 1 distinct linear irreducible characters of U. This is a contradiction. Hence all linear irreducible characters of U appear as irreducible summands of Ind_{C}^{U}(1_{C}), and nowhere else. For the degrees to match, we must have Ind_{C}^{U}(1_{C}) = ∑_{i=1}^{q} ξ_{i}, where the ξ_{i}’s are the (only) q linear irreducible characters of U.

This implies that k = 2, i.e. for any 1_{C} ≠ β_{i} ∈ Irr(C), Ind_{C}^{U}(β_{i}) has exactly 2 irreducible summands in Irr(U). Writing Ind_{C}^{U}(β_{2}) = χ(1)χ+ψ(1)ψ, we get χ(1) = ψ(1) = 2^{n}.

So far, we got that Irr(U) = {ξ_{1}, …, ξ_{q}, χ_{2}, …, χ_{q}, ψ_{2}, …, ψ_{q}}. Now the ξ_{i}’s are real (indeed, for any j ∈ {2, …, q}, ξ_{i}(τ_{j}) = 1_{C}(τ_{j}) = 1, hence ξ_{i}(f_{j}) ∈ {±1}). If all χ_{2}, …, χ_{q} were real, then, since the β_{i}’s are real, χ_{i} + ψ_{i} = Ind_{C}^{U}(β_{i}) is real for all i = 2, …, q, and thus all χ_{2}, …, χ_{q}, ψ_{2}, …, ψ_{q} would be real. But this is not true since (e.g.) f_{2} and f_{2}^{−1} are not conjugate in U. Thus, without loss of generality, χ_{2} is not real and, since {χ_{2}, …, χ_{q}} = {χ_{2}^{a}, a ∈ A}, none of χ_{2}, …, χ_{q} is real.

Now, for all i = 2, …, q, Res_{C}^{U}(\overline{χ_{i}}) = 2^{n}\overline{β_{i}} = 2^{n}β_{i} = Res_{C}^{U}(χ_{i}), so that \overline{χ_{i}} ∈ \{χ_{i}, ψ_{i}\}. From the above remark, we deduce that \overline{χ_{i}} = ψ_{i} for all i = 2, …, q. Finally, the irreducible characters of U are as follows:

\begin{matrix}1_{U} = ξ_{1}, …, ξ_{q} \\ degree 1 \end{matrix}  \begin{matrix}χ_{2}, …, χ_{q}, \overline{χ_{2}}, …, \overline{χ_{q}} \\ degree 2^{n} \end{matrix}

\begin{matrix}ξ_{i} = \overline{ξ_{i}}, i = 1, …, q, \\ Ind_{C}^{U}(1_{C}) = ∑_{i=1}^{q} ξ_{i}. \end{matrix} \begin{matrix}Ind_{C}^{U}(β_{i}) = 2^{n}(χ_{i} + \overline{χ_{i}}), i = 1, …, q, \\ Irr(C) \ \{1_{C}\} = \{β_{2}, …, β_{q}\}. \end{matrix}

Values on 4-elements

Characters of U:

Take any k ∈ {2, …, q} and consider χ_{k}. We have Res_{C}^{U}(χ_{k}) = 2^{n}β_{k}, β_{k} ∈ Irr(C). We define I_{1} = {j ∈ {2, …, q} | β_{k}(τ_{j}) = 1} and I_{−1} = \{j ∈ \{2, …, q\} | β_{k}(τ_{j}) = −1\}. Then |I_{1}| = \frac{q}{2} − 1 and |I_{−1}| = \frac{q}{2}. Take ρ_{k}: U ⟼ GL_{2^{n}}(C) the representation associated to χ_{k}. Using the fact χ_{k}(τ_{j}) = ±2^{n}, depending on if j ∈ I_{1} or j ∈ I_{−1}, we see that Sp(ρ_{k}(τ_{j})) = {1} (the set of eigenvalues of ρ_{k}(τ_{j})) if j ∈ I_{1}, and Sp(ρ_{k}(τ_{j})) = {−1} if j ∈ I_{−1}. Thus

Sp(ρ_{k}(f_{j})) ⊂ \{ \begin{matrix}\{−1, 1\} if j ∈ I_{1} \\ \{−i, i\} if j ∈ I_{−1} \end{matrix}

and

χ_{k}(f_{j}) = \{ \begin{matrix}n_{kj} ∈ Z if j ∈ I_{1} \\ n_{kj}i ∈ Zi if j ∈ I_{−1} \end{matrix} .

Now < χ_{k}, 1_{U} >_{U}= 0 = ∑_{u∈U} χ_{k}(u) = ∑_{u∈C} χ_{k}(u) + ∑_{u∈U\C} χ_{k}(u). But

∑_{u∈C} χ_{k}(u) = |C| < Res_{C}^{U}(χ_{k}), 1_{C} >_{C}= |C| < 2^{n}β_{k}, 1_{C} >_{C}= 0

and

\begin{aligned}∑_{u∈U\C} χ_{k}(u) & = & ∑_{j∈I_{−1}} |Cl_{U}(f_{j})|(χ_{k}(f_{j}) + χ_{k}(f_{j}^{−1})) + ∑_{j∈I_{1}} |Cl_{U}(f_{j})|(χ_{k}(f_{j}) + χ_{k}(f_{j}^{−1})) \\ & = & ∑_{j∈I_{−1}} \frac{q}{2}(n_{kj}i − n_{kj}i) + ∑_{j∈I_{1}} qχ_{k}(f_{j}) \end{aligned}

whence we finally get that

∑_{j∈I_{1}} χ_{k}(f_{j}) = 0 for any 2 ≤ k ≤ q.

(Notice that, even though it is not clearly indicated, the set I_{1} in the above formula depends on k.)

Now, similarly, for any 2 ≤ k ≤ q, ξ_{k} is orthogonal to 1_{U} :

0 = ∑_{u∈U} ξ_{k}(u) = ∑_{u∈C} ξ_{k}(u) + ∑_{u∈U\C} ξ_{k}(u) = |C| + ∑_{j=2}^{q} 2\frac{q}{2}ξ_{k}(f_{j})

whence we get that

∑_{j=2}^{q} ξ_{k}(f_{j}) = −1 for any 2 ≤ k ≤ q.

Induction from U to B:

We write B = Ut_{1} ∪ ⋯ ∪ Ut_{q−1}, where the t_{i}’s are representatives for the cosets of U in B. For any χ ∈ Irr(U) and 2 ≤ k ≤ q, we have (since U ⊲ B = N_{G}(U)) Ind_{U}^{B}(χ)(f_{k}) = ∑_{i=1}^{q−1} χ(t_{i}f_{k}t_{i}^{−1}). Comparing the conjugacy classes of elements of order 4 in U and B, we see that, up to relabelling the f_{j}’s, we can suppose that f_{2}, …, f_{q} are conjugate in B to ρ, and f_{2}^{−1}, …, f_{q}^{−1} are conjugate in B to ρ^{−1}. In particular, f_{k} is conjugate in B to each of f_{2}, …, f_{q}. Since conjugation of f_{k} by elements of a given coset of U in B gives a unique conjugacy class in U, we see that, as i runs through {1, …, q −1}, f_{k}^{Ut_{i}} must run through {Cl_{U}(f_{j}), 2 ≤ j ≤ q}. This implies that

{χ(t_{i}f_{k}t_{i}^{−1}), 1 ≤ i ≤ q − 1} = {χ(f_{j}), 2 ≤ j ≤ q}

and

Ind_{U}^{B}(χ)(f_{k}) = ∑_{j=2}^{q} χ(f_{j}) for all χ ∈ Irr(U), 2 ≤ k ≤ q.

Last step

Now we can finish our description of the irreducible characters of B. Remember only three were missing, μ_{1}, μ_{2} and μ_{3}, respectively induced from U to B by (non-trivial) irreducible characters ν_{1}, ν_{2} and ν_{3} of U of degree 1, 2^{n} and 2^{n}. Thus, w.l.o.g., μ_{1} = Ind_{U}^{B}(ξ_{2}) and μ_{2} = Ind_{U}^{B}(χ_{2}). Thus, writing ρ and ρ^{−1} representatives for the two classes of elements of order 4 in B, we have

μ_{1}(ρ) = ∑_{j=2}^{q} ξ_{2}(f_{j}) = −1

μ_{2}(ρ) = ∑_{j=2}^{q} χ_{2}(f_{j}) = ∑_{j∈I_{1}} χ_{2}(f_{j}) + ∑_{j∈I_{−1}} χ_{2}(f_{j})

which, since ∑_{j∈I_{1}} χ_{2}(f_{j}) = 0, gives μ_{2}(ρ) = ai ∈ Zi. Similarly, μ_{3}(ρ) = bi ∈ Zi. Writing the second orthogonality relation in H for the column of ρ, we get ∑_{μ∈Irr(H)} |μ(ρ)|^{2} = q + a^{2} + b^{2} = |C_{H}(ρ)| = 2q. Thus a^{2} + b^{2} = q = 2^{2n+1}; this forces both a and b to be non-zero. But then μ_{2}(ρ) ∈ Zi \ {0}, so that \overline{μ_{2}} ≠ μ_{2}, and therefore \overline{μ_{2}} = μ_{3}. Furthermore, |a| = |b| = 2^{n}, and, since bi = μ_{3}(ρ) = \overline{μ_{2}(ρ)} = \overline{ai} = −ai, we have b = −a and, up to exchanging μ_{2} and μ_{3}, we have a = 2^{n} and b = −2^{n}.

Remark: A description of the irreducible characters of B, as well as their values on elements of order 4 can actually be found in Suzuki [25].

2.4.3 A generalized perfect isometry

We want to show that there exists a generalized perfect isometry with respect to elements of order (divisible by) 4 between the principal 2-blocks B_{0} and b_{0} of G and B respectively.

First note that, if x ∈ B has order 4, then x ∈ U and C_{B}(x) ≤ U, so that C_{B}(U) ≤ U. Since U is the largest normal 2-subgroup of B, this implies that B has only one 2-block, i.e. b_{0} =Irr(B). Furthermore, the fact that U is a TI set with respect to B implies that the 2-blocks of G have either full defect or defect 0. Since B has only one 2-block, Brauer’s First Main Theorem implies that G has exactly one block of full defect, B_{0}, and any other block has defect 0. Using the notations of Burkhardt [4], we see that G has exactly one character of defect 0, written Π, and thus B_{0} =Irr(G) \ {Π}.

We take from [4] the following fragment of the character table of G (corresponding to characters in B_{0}):

1  f f^{−1}
1_{G}  1  1  1
Ω_{s}, 1 ≤ s ≤ \frac{q−2}{2}  q^{2} + 1  1  1
Θ_{l}, 1 ≤ l ≤ \frac{q+r}{4} (q − r + 1)(q − 1) −1 −1
Λ_{u}, 1 ≤ u ≤ \frac{q−r}{4} (q + r + 1)(q − 1) −1 −1
Γ_{1}  (q − 1)2^{n}  2^{n}i −2^{n}i
Γ_{2}  (q − 1)2^{n}  −2^{n}i 2^{n}i

(Note that |B_{0}| = q + 2 = |Irr(B)| = |b_{0}|.) And, with what we have done before, we can compute the same fragment of the character table of B :

1  ρ ρ^{−1}
1_{B}  1  1  1
φ_{k}, 2 ≤ k ≤ q − 1  1  1  1
μ_{1}  (q − 1) −1 −1
μ_{2}  (q − 1)2^{n} 2^{n}i −2^{n}i
μ_{3} = \overline{μ_{2}}  (q − 1)2^{n} −2^{n}i 2^{n}i

Finally, note that, if x ∈ B has order 4, then C_{G}(x) = C_{B}(x), so that, as

in the previous section, if a bijection with signs between B_{0} and b_{0} preserves the values of characters on 4-elements, it will also preserve contributions. We then see from the two tables above that such a bijection exists; one can consider for example

I: \{ \begin{matrix}1_{G} & ⟼ & 1_{B} \\ Ω_{1} & ⟼ & φ_{2} \\ ⋮ & ⋮ & ⋮ \\ Ω_{(q−2)/2} & ⟼ & φ_{q/2} \\ Θ_{1} & ⟼ & −φ_{q/2+1} \\ ⋮ & ⋮ & ⋮ \\ Θ_{(q+r)/4} & ⟼ & −φ_{(3q+r)/4} \\ Λ_{1} & ⟼ & −φ_{(3q+r)/4+1} \\ ⋮ & ⋮ & ⋮ \\ Λ_{(q−r)/4−1} & ⟼ & −φ_{q−1} \\ Λ_{(q−r)/4} & ⟼ & μ_{1} \\ Γ_{1} & ⟼ & μ_{2} \\ Γ_{2} & ⟼ & μ_{3} \end{matrix}

2.5 Ree Groups

In this section, we consider G = Re(q) = ^{2}G_{2}(q) a Ree group, where q = 3^{2m+1}, for some m ∈ N. We consider U a Sylow 3-subgroup of G and B = N_{G}(U).

2.5.1 The groups G, B and U

The group G is a twisted group of Lie type G_{2}, simple if m ≥ 1, and a description based on a root system of type G_{2} can be found in Carter [7]. In particular, we find in [7] that |G| = q^{3}(q − 1)(q^{3} + 1), so that |U| = q^{3}. Moreover, U can be written U = {x(t, u, v) | t, u, v ∈ F_{q}}, with multiplication given by

x(t_{1}, u_{1}, v_{1})x(t_{2}, u_{2}, v_{2}) = x(t_{1}+t_{2}, u_{1}+u_{2}−t_{1}t_{2}^{3θ}, v_{1}+v_{2}−t_{2}u_{1}+t_{1}t_{2}^{3θ+1}−t_{1}^{2}t_{2}^{3θ})

where θ is the automorphism of F_{q} given by λ^{θ} = λ^{3^{m}} for all λ ∈ F_{q} (and thus 3θ^{2} = 1). We will sometimes write (abusively) θ = 3^{m}.

There exists a subgroup H of B such that B = UH and |H| = q − 1. We can write H = {h(w) , w ∈ F_{q}^{×}} and conjugation of U by H is given by: for any t, u, v, w ∈ F_{q}, w ≠ 0,

h(w)x(t, u, v)h(w)^{−1} = x(w^{2−3θ}t, w^{3θ−1}u, wv).

We find some other structure results about Ree groups in Ward [26]. There can be found the following:

Theorem 2.5. U is disjoint from its conjugates. Its center Z(U) is elementary Abelian of order q. U is of class 3 and contains a normal Abelian subgroup U′ of order q^{2} containing Z(U) which is both the derived group and the Frattini subgroup of U. The members of U \ U′ have order 9, their cubes forming Z(U) \ {1}. We have B = N_{G}(U) = UH where H is cyclic of order q − 1.

The character table of G is also given in this article. G has q+8 conjugacy classes, with representatives 1, R^{a} ≠ 1 for some R of order (q − 1)/2 which is prime to 3, S^{a} ≠ 1 for some S of order (q + 1)/4 which is prime to 3, V and W of order dividing respectively q + 1 − 3^{m+1} and q + 1 + 3^{m+1} (the orders of their centralizers, cf [26], p 85) which are prime to 3, X, T and T^{−1} of order 3 (X ∈ Z(U) and T, T^{−1} ∈ U \ Z(U), cf [26], pp 78-80), Y, Y T and Y T^{−1} of order 9 (cf [26], p 82), J of order 2, JT and JT^{−1} which belong to groups whose order has 3-valuation 1 (cf [26], p 83) so which order is not divisible by 9, and JR^{A} ≠ J and JS^{a} ≠ J of respective orders o(J).o(R^{a}) and o(J).o(S^{a}) which are not divisible by 3.

Note that, following Carter, we write q = 3^{2m+1}, while Ward writes q = 3^{2k+1} and m = 3^{k}. The m in Ward’s paper therefore corresponds to 3^{m} in our notations.

The number and lengths of the conjugacy classes of U, as well as the number and degrees of the irreducible characters of U and B can be found in Eaton [11]. However, in order to obtain the values we need, we must compute some details Eaton doesn’t give about the irreducible characters of B. We therefore compute again everything explicitly.

2.5.2 Conjugacy classes of U

Conjugation in U, centralizers

We see from the formula for multiplication in U that the identity element of U is x(0, 0, 0), and that, since the center Z(U) of U has order q, it can be identified with {x(0, 0, v) , v ∈ F_{q}}. We also obtain informations about the order of elements : x(t, u, v)^{2} = x(−t, −u − t^{3θ+1}, −v − tu) and x(t, u, v)^{3} = x(0, 0, −t^{3θ+2}), so that x(t, u, v)^{9} = 1, and x(t, u, v) has order 3 if and only if t = 0. Moreover, as gcd(3^{m+1} − 1, 3θ + 2) = 1, we have {−t^{3θ+2} , t ∈ F_{q}^{×}} = F_{q}^{×}, so that the cubes of the elements of order 9 of U are Z(U) \ {1} = {x(0, 0, v) , v ∈ F_{q}^{×}}.

Furthermore, the inverse of an element x(t, u, v) of U is

x(t, u, v)^{−1} = x(−t, −u − t^{3θ+1}, −v − tu + t^{3θ+2}).

Hence conjugation in U is given by

x(t_{1}, u_{1}, v_{1})x(t_{2}, u_{2}, v_{2})x(t_{1}, u_{1}, v_{1})^{−1} =
x(t_{2}, u_{2} − t_{2}^{3θ}t_{1} + t_{2}t_{1}^{3θ}, v_{2} − t_{2}u_{1} + t_{1}u_{2} + t_{2}^{2}t_{1}^{3θ} − 2t_{2}^{3θ}t_{1}^{2} + t_{2}^{3θ+1}t_{1}).

In particular, conjugation in U preserves the first “coordinate”.

Now take any t_{1}, t_{2}, u_{1}, u_{2}, v_{1}, v_{2} ∈ F_{q}. Then x(t_{1}, u_{1}, v_{1})x(t_{2}, u_{2}, v_{2}) = x(t_{2}, u_{2}, v_{2})x(t_{1}, u_{1}, v_{1}) if and only if

\{ \begin{matrix}t_{1} + t_{2} = t_{2} + t_{1} \\ t_{1}t_{2}^{3θ} = t_{2}t_{1}^{3θ} \\ t_{2}u_{1} − t_{1}t_{2}^{3θ+1} + t_{1}^{2}t_{2}^{3θ} = t_{1}u_{2} − t_{2}t_{1}^{3θ+1} + t_{2}^{2}t_{1}^{3θ} (†) \end{matrix}

Writing ω = t_{1}t_{2}^{3θ} = t_{2}t_{1}^{3θ}, we get that (†) ⟺ t_{1}(u_{2} − 2ω) = t_{2}(u_{1} − 2ω) (†′).

If t_{1}, u_{1} and v_{1} are now fixed, we can solve the above system of equations in t_{2}, u_{2}, v_{2} ∈ F_{q}, and we eventually obtain: C_{U}(x(0, u_{1} ≠ 0, v_{1})) = {x(0, u_{2}, v_{2}) | u_{2}, v_{2} ∈ F_{q}} has order q^{2}, and thus Cl_{U}(x(0, u_{1} ≠ 0, v_{1})) has length q. C_{U}(x(t ≠ 0, u, v_{1})) = {x(0, 0, v_{2}) , x(t, u, v_{2}) , x(−t, −t^{3θ+1} − u, v_{2}) | v_{2} ∈ F_{q}} has order 3q, and Cl_{U}(x(t ≠ 0, u, v_{1})) has length q^{2}/3.

Using the formula for conjugation in U and the lengths we found, we see that, given t, u, v ∈ F_{q}, Cl_{U}(x(0, u, v)) = {x(0, u, v + t_{1}u) , t_{1} ∈ F_{q}}, so that, if u ≠ 0, then Cl_{U}(x(0, u, v)) = {x(0, u, v_{2}) , v_{2} ∈ F_{q}}, giving us q − 1 such classes, uniquely determined by u. Moreover, x(t, u_{2}, v_{2})x(t, u, v)x(t, u_{2}, v_{2})^{−1} = x(t, u, v + t(u − u_{2})), so that, if t ≠ 0, then Cl_{U}(x(t, u, v)) ⊃ {x(t, u, v_{2}) , v_{2} ∈ F_{q}}. Hence, by conjugation, the first coordinate is fixed and the last can be changed to any one without modifying the second one. We obtain that x(t, u_{2}, v_{2}) ∈ Cl_{U}(x(t, u, v)) if and only if there exists t_{2} ∈ F_{q} such that u_{2} = u − t_{2}t^{3θ} + tt_{2}^{3θ}. We can show that there are q/3 such u_{2}’s. Indeed, take any t_{2}, t_{3} ∈ F_{q}, and the associated u_{2} and u_{3}. Then,

u_{2} = u_{3} ⟺ tt_{2}^{3θ} − t_{2}t^{3θ} = tt_{3}^{3θ} − t_{3}t^{3θ}
⟺ t(t_{2}^{3θ} − t_{3}^{3θ}) = (t_{2} − t_{3})t^{3θ}
⟺ t(t_{2} − t_{3})^{3θ} = (t_{2} − t_{3})t^{3θ}

(this last equivalence being true since the caracteristic is 3 and (−1)^{3θ} = −1). And this has 3 solutions, t_{2} − t_{3} ∈ {0, ±t}. Hence we obtain |F_{q}|/3 = q/3 distinct values for u_{2}. Thus, we get

Cl_{U}(x(t ≠ 0, u, v)) = {x(t, u − t_{2}t^{3θ} + tt_{2}^{3θ}, v_{2}) | t_{2}, v_{2} ∈ F_{q}},

giving us 3(q − 1) such classes, determined by t and 3 distinct values u_{1}, u_{2} and u_{3} ∈ F_{q}.

This gives us the whole of U :

• q classes of length 1, type x(0, 0, v), parametrized by v ∈ F_{q}

• q − 1 classes of length q, type x(0, u ≠ 0, v), parametrized by u ∈ F_{q}^{×}

• 3(q − 1) classes of length q^{2}/3, type x(t ≠ 0, u, v), parametrized by t ∈ F_{q}^{×} and 3 values of u ∈ F_{q}.

Thus U has 5q − 4 conjugacy classes.

H-conjugacy classes of U

First, recall that, for any t, u, v, w ∈ F_{q}, w ≠ 0,

h(w)x(t, u, v)h(w)^{−1} = x(w^{2−3θ}t, w^{3θ−1}u, wv),

so that conjugation by elements of H preserves the three types of conjugacy classes of U we just described.

Now, H acts by conjugation on the conjugacy classes of U. We find the number and sizes of the orbits under this action.

• Take v ∈ F_{q}. If v ≠ 0, then {wv , w ∈ F_{q}^{×}} = F_{q}^{×}, so that we get two H-orbits : {x(0, 0, 0)} and Z(U) \ {1} = {x(0, 0, v) , v ∈ F_{q}^{×}}.

• Take u ∈ F_{q}^{×} and w ∈ F_{q}^{×}. Then

h(w)Cl_{U}(x(0, u, v))h(w)^{−1} = Cl_{U}(x(0, u, v)) ⟺ w^{3θ−1}u = u

and this has 2 solutions w ∈ F_{q}^{×} (w = ±1). This gives us two H-orbits, of size (q − 1)/2.

• Take t ∈ F_{q}^{×} and w ∈ F_{q}^{×}. Then

h(w)Cl_{U}(x(t, u, v))h(w)^{−1} = Cl_{U}(x(t, u, v)) ⟹ w^{2−3θ}t = t

Now w^{2−3θ}t = t ⇔ w^{3θ−2} = 1 ⇔ w^{3^{m+1}−2} = 1. However, we can see that gcd(3^{2m+1} − 1, 3^{m+1} − 2) = 1, and thus w^{3^{m+1}−2} = 1 has a unique solution w ∈ F_{q} (w = 1). This gives us three H-orbits, of size q − 1.

Finally, there are 7 H-orbits on the conjugacy classes of U.

2.5.3 Irreducible characters of U

Actions of B

If L = U or Z(U), then L ⊲ B, and B acts by conjugation on the conjugacy classes of L (via x^{b} = b^{−1}xb) and on Irr(L) (via χ^{b}(x) = χ(bxb^{−1})). Corollary (11.10) in Curtis-Reiner [9] implies that the number of orbits in these two actions are the same. Furthermore, since U is normal in B = UH and U acts trivially on the conjugacy classes of Z(U) and of U, the B-orbits in these actions are in fact the H-orbits. The study of H-orbits we made before shows that there are 2 H-orbits in Z(U), and thus in Irr(Z(U)), and 7 H-orbits in Cl(U), and thus in Irr(U).

Induction from Z(U) to U

We write C = Z(U). Since {1_{C}} is clearly one H-orbit of Irr(C), we see that H must act transitively on Irr(C) \ {1_{C}}, and Clifford’s Theorem gives us an injective map

Irr(C) ⟶ Irr(U)
β_{i} ⟼ χ s.t. < Ind_{C}^{U}(β_{i}), χ >_{U}≠ 0

where Irr(C) = {β_{1} = 1_{C}, β_{2}, …, β_{q}}. Furthermore, since |H| = |Irr(C) \ {1_{C}}|, we have, as in Suzuki groups,

|Irr(U)| = |{Irred. summ. of Ind_{C}^{U}(1_{C})}| + (q − 1)k

where k is the (common) number of irreducible summands of any Ind_{C}^{U}(β_{i}), i ≥ 2.

We can write Ind_{C}^{U}(1_{C}) = ∑_{χ∈I_{1}⊂Irr(U)} χ(1)χ. Now 1_{U} appears with multiplicity 1 in Ind_{C}^{U}(1_{C}), and so does any linear irreducible summand of Ind_{C}^{U}(1_{C}). Since Ind_{C}^{U}(1_{C})(1) = [U : C] = q^{2} is a power of 3, and since any non-linear irreducible summand of Ind_{C}^{U}(1_{C}) has degree a non-trivial power of 3, we see that there must be (at least) 2 other linear irreducible summands in Ind_{C}^{U}(1_{C}), α and β say. Supposing some Ind_{C}^{U}(β_{i}) (β_{i} ≠ 1_{C}) has a linear irreducible summand, we would get an orbit of q − 1 distinct linear irreducible characters of U, all distinct from 1_{U}, α and β (since, by Clifford, 1_{U}|_{C} = α|_{C} = β|_{C} = 1_{C}), giving us q+2 linear irreducible characters of U. This is a contradiction, since [U : U′] = q. Hence all linear irreducible characters of U appear as irreducible summands of Ind_{C}^{U}(1_{C}) and nowhere else. Writing Ind_{C}^{U}(1_{C}) = ∑_{i=1}^{q} λ_{i} + ∑_{i∈I} χ_{i}(1)χ_{i}, where the λ_{i}’s are linear, we have |Irr(U)| = 5q − 4 = q + |I| + (q − 1)k. Thus 4q − 4 = |I| + (q − 1)k, and k ≤ 4.

k = 4 implies I = ∅ and Ind_{C}^{U}(1_{C}) = ∑_{i=1}^{q} λ_{i}, which is impossible since the left has degree q^{2} and the right has degree q.

Suppose k = 2. Then we can write (using Clifford) Ind_{C}^{U}(β_{2}) = χ(1)χ + ψ(1)ψ and (degrees) 3^{2a} = q^{2} = χ(1)^{2} +ψ(1)^{2} = 3^{2b} +3^{2c}. Then b = c implies 2|3^{2a}, which is impossible, and b < c implies 3^{2a} = 3^{2b}(1 + 3^{2(c−b)}), which is even, so this is impossible too.

Suppose k = 3. Then, q^{2} = χ(1)^{2} + ψ(1)^{2} + η(1)^{2}, which can similarly be shown to be impossible.

We deduce that k = 1. Hence there exist ψ_{2}, …, ψ_{q} ∈ Irr(U) such that

Ind_{C}^{U}(β_{i}) = ψ_{i}(1)ψ_{i} for i = 2, …, q

and thus ψ_{i}(1) = q for i = 2, …, q.

Furthermore, |I| = 3q − 3, Ind_{C}^{U}(1_{C}) = ∑_{i=1}^{q} λ_{i} + ∑_{i=1}^{3q−3} χ_{i}(1)χ_{i}, and, by Clifford’s Theorem, the χ_{i}’s have C = Z(U) in their kernel, so are in fact irreducible characters of the quotient U/Z(U).

The quotient V = U/Z(U)

From the central series {1} ⊲ Z(U) ⊲ U′ ⊲ U, we see that V = U/Z(U) has order q^{2}, and has a normal Abelian subgroup (U′/Z(U)) of order q such that the corresponding quotient is Abelian of order q.

We have seen that Z(U) = {x(0, 0, v) , v ∈ F_{q}}, and, from the formula for multiplication in U, we see that U/Z(U) can be parametrized by {y(t, u) | t, u ∈ F_{q}}, where multiplication is given by

y(t_{1}, u_{1})y(t_{2}, u_{2}) = y(t_{1} + t_{2}, u_{1} + u_{2} − t_{1}t_{2}^{3θ}).

We have 1_{U/Z(U)} = y(0, 0), y(t, u)^{−1} = y(−t, −u − t^{3θ+1}), and

y(t_{1}, u_{1})y(t_{2}, u_{2})y(t_{1}, u_{1})^{−1} = y(t_{2}, u_{2} − t_{1}t_{2}^{3θ} + t_{2}t_{1}^{3θ}).

From the results in U, we see that the conjugacy classes of V are as follows :

• {y(0, 0)}

• q − 1 classes of length 1 : {y(0, u)} , u ∈ F_{q}^{×}

• 3(q − 1) classes of length q/3, of type y(t, u) for t ∈ F_{q}^{×} and 3 values of u ∈ F_{q}.

The subgroup W = {y(0, u) , u ∈ F_{q}} is both the center and the derived group of V. Thus V has q linear irreducible characters and 3q − 3 nonlinear irreducible characters, the ones we are looking for, χ_{1}, …, χ_{3q−3}. As V has odd order q^{2}, each non-trivial irreducible character of V is non-real. Furthermore, by Clifford’s Theorem, every irreducible character of V appears as irreducible summand of some character induced from W, with multiplicity its degree.

Irreducible characters of W = Z(V):

We have V = U/Z(U) ⊲ B/Z(U) = (U/Z(U))H. Thus B/Z(U) acts by conjugation on Z(V) = W, and this action reduces to an action of H. From the study of H-classes of U, we see that there are 3 orbits under this action, giving us 3 orbits under the derived action of B/Z(U) on Irr(W) = {γ_{1} = 1_{W} , γ_{2}, …, γ_{q}}, where the γ_{i}’s are linear. Since {1_{W} } is one H-orbit, {γ_{2}, …, γ_{q}} must be a union of 2 H-orbits. Now the length of an orbit divides the order of H, that is q − 1, and this forces each of these 2 orbits to have length (q − 1)/2. Now q −1 ≡ (−1)^{2m+1} −1 ≡ −2 (mod 4), so that (q −1)/2 is odd. Moreover, W has odd order q. Thus the only real irreducible character of W is 1_{W}, and γ_{i} ≠ \overline{γ_{i}} for all i = 2, …, q. Supposing that \overline{γ_{2}} belongs to the same H-orbit as γ_{2}, \overline{γ_{2}} = γ_{2}^{h} say, and taking any γ_{i} in this orbit, it is easy to see (using the definitions of the actions of H) that then γ_{i}^{h} = \overline{γ_{i}}, so that \overline{γ_{i}} belongs to the same H-orbit. The orbit of γ_{2} would then have even length, which is a contradiction. This shows that, up to relabelling the γ_{i}’s, the H-orbits of Irr(W) are \{1_{W} \}, \{γ_{2}, … γ_{(q+1)/2}\} and \{\overline{γ_{2}}, … \overline{γ_{(q+1)/2}}\}.

Induction from W to V:

If γ_{i} and γ_{j} belong to the same H-orbit of Irr(W), then Ind_{W}^{V} (γ_{i}) and Ind_{W}^{V} (γ_{j}) have the same number of irreducible components, with the same degrees (that is, we can build a bijection preserving the degree between the sets of irreducible components of Ind_{W}^{V} (γ_{i}) and Ind_{W}^{V} (γ_{j})). Furthermore, complex conjugation gives the same kind of bijection between irreducible components of Ind_{W}^{V} (γ_{i}) and Ind_{W}^{V} (\overline{γ_{i}}), whence all the Ind_{W}^{V} (γ_{i})’s, 2 ≤ i ≤ q, have the same number of irreducible components, with the same degrees. Moreover, the subsets of Irr(V) obtained in this way are disjoint.

Now 1_{V} appears with multiplicity 1 in Ind_{W}^{V} (1_{W}), which has degree q. Hence at least two other linear characters of V must appear (each with multiplicity 1). Supposing a linear character appears in some Ind_{W}^{V} (γ_{i}), i ≥ 2, we would get one in each Ind_{W}^{V} (γ_{j}), j ≥ 2, giving us (at least) q + 2 distinct linear characters of V, which is impossible. Hence each linear character of V appears in Ind_{W}^{V} (1_{W}) and nowhere else. Comparing degrees, we see that Ind_{W}^{V} (1_{W}) = ∑_{i=1}^{q} μ_{i}, where the μ_{i}’s are (all) the irreducible linear characters of V. Writing k′ the number of irreducible components of any Ind_{W}^{V} (γ_{i}), i ≥ 2, we have |Irr(V)| = 4q − 3 = q + k′(q − 1), so that k′ = 3.

Writing Ind_{W}^{V} (γ_{2}) = χ_{1}(1)χ_{1} + χ_{2}(1)χ_{2} + χ_{3}(1)χ_{3}, we obtain q = 3^{2m+1} = χ_{1}(1)^{2} + χ_{2}(1)^{2} + χ_{3}(1)^{2}, which implies χ_{1}(1) = χ_{2}(1) = χ_{3}(1) = 3^{m}.

Finally, V has q linear irreducible characters, and 3q − 3 irreducible characters of degree 3^{m}.

Irreducible characters of U

We summarize the results we obtained so far about the character table of U. By convention, in character tables, we will take the first line to correspond to the values on 1, and the first column to correspond to the trivial character.

1_{U} = λ_{1}, …, λ_{q}

Irr(U/U′) degree 1

\begin{matrix}ψ_{2}, …, ψ_{q} \\ degree q \end{matrix}

x(0, 0, v), v ∈ F_{q} length 1 order 3 1

\begin{matrix}qψ_{i} = Ind_{Z(U)}^{U}(β_{i}) \\ β_{i} ∈ Irr(Z(U)) \ \{1_{Z(U)}\} \\ Z(U) ≅ (F_{q}, +) \end{matrix}

x(0, u, v), u ∈ F_{q}^{×} length q order 3 1

0

x(t, u_{i}, v), t ∈ F_{q}^{×}, i = 1, 2, 3 length q^{2}/3 order 9 three times each non-first line of the character table of U/U′ Same line ⇔ same t U/U′ = {x(t), t ∈ F_{q}} U/U′ ≅ (F_{q}, +)

0

χ_{1}, …, χ_{3q−3}

irreducible characters of U/Z(U) degree 3^{m}

x(0, 0, v), v ∈ F_{q}

length 1

order 3 3^{m}

x(0, u, v), u ∈ F_{q}^{×} length q order 3

3^{m}. ( \begin{matrix}three times each non-first column of the \\ character table of W = Z(U/Z(U)) \\ Same column ⇔ χ_{i}, χ_{j} belong \\ to the same Ind_{W}^{V} (γ_{k}) \\ W = \{x(0, u), u ∈ F_{q}\} ≅ (F_{q}, +) \end{matrix} )

x(t, u_{i}, v), t ∈ F_{q}^{×}, i = 1, 2, 3 length q^{2}/3 order 9

???????

H-orbits of Irr(U)

We have seen that there are two orbits in the actions of H by conjugation on Irr(Z(U)) and Z(U). The orbits of Irr(Z(U)) must be {1_{Z(U)}} and {β_{2}, …, β_{q}}. Now H acts by conjugation on Irr(U) and on Cl(U), and the underlying action on Cl_{U}(Z(U)) = Z(U) is the one before. Via Clifford’s Theorem, we have that, for h ∈ H, ψ ∈ Irr(U) and 2 ≤ i ≤ q,

< ψ, Ind_{Z(U)}^{U}(β_{i}) >_{U}≠ 0 ⟺< ψ^{h}, Ind_{Z(U)}^{U}(β_{i}^{h}) >_{U}≠ 0

(indeed, both are equal to < ψ|_{Z(U)}, β_{i} >_{Z(U)}=< ψ^{h}|_{Z(U)}, β_{i}^{h} >_{Z(U)}= ψ(1) = ψ^{h}(1)). As Ind_{Z(U)}^{U}(β_{i}) = qψ_{i} for each i = 2, …, q, we see that the ψ_{i}’s form one H-orbit of Irr(U). {1_{U}} is another one. However, we know that Irr(U) has 7 H-orbits. Thus {λ_{2}, …, λ_{q}, χ_{1}, …, χ_{3q−3}} is a union of 5 H-orbits. Furthermore, because of the preservation of the degree under the action of H, {λ_{i}, i ≥ 2} and {χ_{i}, 1 ≤ i ≤ 3q − 3} are unions of H-orbits.

Now the χ_{i}’s are characters of V = U/Z(U). They are all non-real, and can be written as \{χ_{1}, …, χ_{(3q−3)/2}, \overline{χ_{1}}, …, \overline{χ_{(3q−3)/2}}\}, the separation between the χ_{i}’s and the \overline{χ_{i}}’s corresponding to the one we introduced between the γ_{i}’s and the \overline{γ_{i}}’s respectively. For γ_{i} ∈ Irr(Z(V)), i ≥ 2, and χ ∈ Irr(V), we have

< χ, Ind_{Z(V)}^{V}(γ_{i}) >_{V} =< \overline{χ}, \overline{Ind_{Z(V)}^{V}(γ_{i})} >_{V} .

Since \{γ_{2}, …, γ_{(q+1)/2}\} and \{\overline{γ_{2}}, …, \overline{γ_{(q+1)/2}}\} are two H-orbits, this shows that χ and \overline{χ} have distinct orbits. Furthermore, for χ, ψ ∈ Irr(V) and h ∈ H, we have

< χ^{h}, ψ^{h} >_{V} =< χ, ψ >_{V} =< \overline{χ}, \overline{ψ} >_{V} =< \overline{χ}^{h}, \overline{ψ}^{h} >_{V}

and \overline{χ}^{h} = \overline{χ^{h}}. Hence the orbits of χ and \overline{χ} have the same length and consist of 2 by 2 conjugate characters. Moreover, since < χ^{h}|_{Z(V)}, γ_{i}^{h} >_{Z(V)}= < χ|_{Z(V)}, γ_{i} >_{Z(V)}, we see that \{χ_{1}, …, χ_{(3q−3)/2}\} and \{\overline{χ_{1}}, …, \overline{χ_{(3q−3)/2}}\} are unions of H-orbits, and that there is a bijection preserving the length between the two corresponding sets of orbits (namely, complex conjugation). Moreover, the orbits have size at least (q − 1)/2, which is the size of γ_{i}^{H}.

This shows that {χ_{i}, 1 ≤ i ≤ 3q − 3} is a union of an even number of H-orbits, which is strictly less than 5, and must therefore be 2 or 4. If it were 2, these orbits would have length (3q − 3)/2 > q − 1 = |H|, which is impossible. Thus, there are 4 orbits, of 2 lengths l and l′, and we can write 3q − 3 = 2l + 2l′, with l, l′ ≥ (q − 1)/2. Also l, l′ | |H| = q − 1, so that l, l′ ∈ {(q −1)/2, q −1}, and this implies that l = (q −1)/2 say, and l′ = q −1.

Finally, the λ_{i}’s must form one H-orbit, and, relabelling the χ′_{i}s, we have the following 7 H-orbits of Irr(U) :

{1_{U}} , {λ_{2}, …, λ_{q}} , {ψ_{2}, …, ψ_{q}}
{χ_{i,j}, i = 1, 2 and 1 ≤ j ≤ (q − 1)/2} , {χ_{3,j}, 1 ≤ j ≤ (q − 1)/2}
\{\overline{χ_{i,j}}, i = 1, 2 and 1 ≤ j ≤ (q − 1)/2\} , \{\overline{χ_{3,j}}, 1 ≤ j ≤ (q − 1)/2\}

where Ind_{Z(V)}^{V}(γ_{j+1}) = 3^{m}(χ_{1,j} + χ_{2,j} + χ_{3,j}) for 1 ≤ j ≤ (q − 1)/2.

2.5.4 Irreducible characters of B = N_{G}(U)

Clifford’s Theory

The following description of Clifford’s Theory can be found in Isaacs [16]. Recall that B acts on Irr(U) like H does. Take any η ∈ Irr(B). If, for some χ ∈ Irr(U), < η, Ind_{U}^{B}(χ) >_{B}=< η|_{U}, χ >_{U}≠ 0, then η|_{U} = e ∑_{i=1}^{t} χ^{h_{i}}, where e =< η|_{U}, χ >_{U}, t = [B : I_{B}(χ)] with I_{B}(χ) the inertial subgroup of χ, and {χ^{h_{i}}; i = 1 … t} is the H-orbit of χ.

Now take any θ ∈ Irr(U), and write Ind_{U}^{B}(θ) = ∑_{i∈I} e_{i}μ_{i}, with μ_{i} ∈ Irr(B) for i ∈ I and μ_{i} ≠ μ_{j} if i ≠ j ∈ I (and thus, as above, μ_{i}|_{U} = e_{i} ∑_{h_{j}} θ^{h_{j}}). Writing T = I_{B}(θ), we then have Ind_{U}^{T}(θ) = ∑_{i∈I} e_{i}ν_{i}, with 0 ≠ e_{i} ∈ N and ν_{i} ∈ Irr(T); moreover, for i ∈ I, μ_{i} = Ind_{T}^{B}(ν_{i}) and thus μ_{i}(1) = [B : T]ν_{i}(1).

To study the e_{i}’s, it suffices to restrict our attention to T = I_{B}(θ), U ⊲ T and θ is T -invariant. We have Ind_{U}^{T}(θ) = ∑_{i∈I} e_{i}ν_{i}, and ν_{i}|_{U} = e_{i}θ (since the T -orbit of θ is just {θ}). Hence ν_{i}(1) = e_{i}θ(1), and

[T : U]θ(1) = Ind_{U}^{T}(θ)(1) = ∑_{i∈I} e_{i}ν_{i}(1) = ∑_{i∈I} e_{i}^{2}θ(1).

Thus ∑_{i∈I} e_{i}^{2} = [T : U].

Irreducible characters of B

We now apply the previous results to the irreducible characters of U, and obtain all the irreducible characters of B.

If θ ∈ \{λ_{2}, …, λ_{q}, ψ_{2}, …, ψ_{q}, χ_{1,j}, χ_{2,j}, \overline{χ_{1,j}}, \overline{χ_{2,j}}\}, then the H-orbit of θ has length q − 1 = |H|, so that I_{H}(θ) = {1} and I_{B}(θ) = U. Hence Ind_{U}^{B}(θ) is irreducible, of degree (q − 1)θ(1).

If θ ∈ \{χ_{3,j}, \overline{χ_{3,j}} | 1 ≤ j ≤ (q − 1)/2\}, then the H-orbit of θ has length (q − 1)/2, so that |I_{H}(θ)| = 2, and thus I_{H}(θ) =< J > where J is the involution of H, and T = I_{B}(θ) = U < J >. Hence Ind_{U}^{T}(θ) = ∑_{i∈I} e_{i}ν_{i} and ∑_{i∈I} e_{i}^{2} = [T : U] = 2, so that |I| = 2 and e_{1} = e_{2} = 1. Thus Ind_{U}^{T}(θ) is the sum of two irreducible characters ν_{1} and ν_{2} of T, each with multiplicity 1, and of same degree as θ. Then Ind_{U}^{B}(θ) = μ_{1} + μ_{2}, with μ_{1}, μ_{2} ∈ Irr(B) distinct, and μ_{1}(1) = μ_{2}(1) = [B : T]ν_{1,2}(1) = θ(1)(q − 1)/2.

Furthermore, two characters of the same H-orbit of Irr(U) give the same induced character of B (and thus the same subset or Irr(B)), and two distinct orbits give two disjoint subsets of Irr(B). We obtain

Irr(U)  Irr(B)
{λ_{2}, …, λ_{q}} ⟶ λ, degree q − 1
{ψ_{2}, …, ψ_{q}}  ⟶ ψ, degree (q − 1)q
\{χ_{i,j}, i = 1, 2 and 1 ≤ j ≤ \frac{q−1}{2}\} ⟶ χ, degree (q − 1)3^{m}
\{\overline{χ_{i,j}}, i = 1, 2 and 1 ≤ j ≤ \frac{q−1}{2}\} ⟶ \overline{χ}, degree (q − 1)3^{m}
\{χ_{3,j}, 1 ≤ j ≤ \frac{q−1}{2}\}  ⟶ μ_{1}, μ_{2}, degree \frac{q−1}{2}3^{m}
\{\overline{χ_{3,j}}, 1 ≤ j ≤ \frac{q−1}{2}\}  ⟶ \overline{μ_{1}}, \overline{μ_{2}}, degree \frac{q−1}{2}3^{m}

Finally, B has (at least) q − 1 linear irreducible characters α_{1} = 1_{B}, … α_{q−1} given by α_{i} = π ∘α̃_{i}, where π: B = UH ⟶ H is the natural homomorphism and Irr(H) = {α̃_{i}, 1 ≤ i ≤ q − 1}. Each of the α_{i}’s appears with multiplicity 1 in Ind_{U}^{B}(1_{U}), and, since Ind_{U}^{B}(1_{U})(1) = q − 1, we have Ind_{U}^{B}(1_{U}) = ∑_{i=1}^{q−1} α_{i}, and the α_{i}’s are the only linear irreducible characters of B.

Values on elements of order 9

From the results on H-classes of U, we see that there are three classes of elements of order 9 (of type x(t ≠ 0, u_{i}, v)) in B. We take representatives x_{1}, x_{2} and x_{3}, and we want the values of Irr(B) on the x_{i}’s.

The α_{i}’s are lifted from H, so have U in their kernel, so that α_{i}(x_{j}) = 1 for all i = 1, …, q − 1 and j = 1, 2, 3.

We have ψ|_{U} = ∑_{i=2}^{q} ψ_{i}, and the ψ_{i}’s are 0 on U \ Z(U). Thus ψ(x_{j}) = 0 for all j = 1, 2, 3.

We have λ|_{U} = ∑_{i=2}^{q} λ_{i}. For each j = 1, 2, 3, using the fragment of the character table of U we found, the second orthogonality relation applied (in U/U′) to x_{j} and 1 gives ∑_{i=1}^{q} λ_{i}(x_{j})λ_{i}(1) = 0, and λ_{i}(1) = 1 for all i, so that λ(x_{j}) = −λ_{1}(x_{j}) = −1 (note that this doesn’t depend on the U-conjugacy class in which we took x_{j}).

Now we have (χ + μ_{1})|_{U} = ∑_{k=1}^{(q−1)/2}(χ_{1,k} + χ_{2,k} + χ_{3,k}), a character of U/Z(U), and

3^{m}(χ + μ_{1})|_{U/Z(U)} = ∑_{k=1}^{\frac{q−1}{2}} 3^{m}(χ_{1,k} + χ_{2,k} + χ_{3,k}) = ∑_{k=1}^{\frac{q−1}{2}} Ind_{W}^{U/Z(U)}(γ_{k+1}),

where Irr(W) = {γ_{1} = 1_{W} , γ_{2}, …, γ_{q}}, and, since W = Z(U/Z(U)), 3^{m}(χ + μ_{1})|_{U} ≡ 0 on (U/Z(U)) \ W. Hence (χ + μ_{1})(x_{j}) = 0 for j = 1, 2, 3. Since μ_{1}|_{U} = μ_{2}|_{U}, we have, for some a, b, c ∈ C,

χ μ_{1}, μ_{2} \overline{χ} \overline{μ_{1}}, \overline{μ_{2}} α_{1}, …, α_{q−1} λ ψ
x_{1} a −a ā −ā  1  −1 0
x_{2} b −b b̄ −b̄  1  −1 0
x_{3} c −c c̄ −c̄  1  −1 0

The second orthogonality relation applied to each line gives, since |C_{B}(x_{i})| = 3q, |a| = |b| = |c| = 3^{m}.

Now we have, for any t, u, v ∈ F_{q}, h(−1)x(t, u, v)h(−1)^{−1} = x(−t, u, −v). If we take u = t = 1, then x(t, u, v)^{−1} = x(−1, 1, −v) = x(−t, u, −v). Hence x(1, 1, v) and x(1, 1, v)^{−1} are conjugate in B, to x_{1} say. Thus a is real, and a = ±3^{m}.

Now the second orthogonality relation applied to the first and second line and to the first and third line gives that Re(b) =Re(c) = ∓3^{m}/2. Together with |b| = |c| = 3^{m}, this gives us that b, c ∉ R, so that c = b̄ (and x_{2}^{−1} is conjugate to x_{3}), and, writing ω = e^{2iπ/3}, we have b3^{−m} ∈ {ω, ω̄, −ω, −ω̄}. Hence, we see that, up to exchanging χ and \overline{χ} (and thus μ_{i} and \overline{μ_{i}}), or x_{2} and x_{3}, there exists ε ∈ {±1} such that a = ε3^{m}, b = ε3^{m}ω̄ and c = ε3^{m}ω.

2.5.5 A generalized perfect isometry

Since the Sylow 3-subgroup U of G is disjoint from its conjugates, the 3-blocks of G have either full defect or defect 0. Furthermore, we have seen that the elements of Z(U) \ {1} form one H-orbit. Since H = q − 1 = |Z(U) \ {1}|, this shows that each element of Z(U) \ {1} is centralized by no non-identity element of H. Thus we get that, for z ∈ Z(U) \ {1}, C_{B}(z) = U, so that C_{B}(U) ≤ U, and, as U = O_{3}(B), B has only one 3-block, the principal one b_{0}. This implies that G has only one block B_{0} of maximal defect. Since G has only one character ξ_{3} of defect 0, we get b_{0} = Irr(B) and B_{0} = Irr(G) \ {ξ_{3}}. In particular, we see that |B_{0}| = |b_{0}| = q + 7.

In G, the only elements of order divisible by 9 have order precisely 9. Hence the same must be true in B. We have representatives for the 9-singular classes Y, Y T and Y T^{−1} in G, and x_{1}, x_{2} and x_{3} in B. We have already seen that |C_{B}(x_{i})| = 3q for i = 1, 2, 3. Using the character table of G in [26], we find that |C_{G}(Y)| = |C_{G}(Y T)| = |C_{G}(Y T^{−1})| = 3q. Here again, it is thus sufficient to find a bijection with signs between B_{0} and b_{0} wich preserves the values of characters on 9-singular elements.

We now consider the following fragments of character table for B_{0} (cf [26]) and b_{0}:

ξ_{4} ξ_{1}, ξ_{2}, η_{r}, η′_{r} η_{t}, η′_{t}, η_{i}^{−}, η_{i}^{+} ξ_{5}, ξ_{6} ξ_{7}, ξ_{8} ξ_{9}  ξ_{10}
Y  0  1  −1  3^{m} 3^{m} −3^{m} −3^{m}
Y T 0  1  −1  3^{m}ω̄ 3^{m}ω −3^{m}ω̄ −3^{m}ω
Y T^{−1} 0  1  −1  3^{m}ω 3^{m}ω̄ −3^{m}ω −3^{m}ω̄

(where the numbers of exceptionnal characters are |{η_{r}, η′_{r}}| = (q − 3)/2 and |{η_{t}, η′_{t}, η_{i}^{−}, η_{i}^{+}}| = (q − 1)/2)

ψ α_{1}, …, α_{q−1} λ μ_{1}, μ_{2}  \overline{μ_{1}}, \overline{μ_{2}}  χ  \overline{χ}
x_{1} 0  1  −1 −ε3^{m} −ε3^{m} ε3^{m} ε3^{m}
x_{2} 0  1  −1 −ε3^{m}ω̄ −ε3^{m}ω ε3^{m}ω̄ ε3^{m}ω
x_{3} 0  1  −1 −ε3^{m}ω −ε3^{m}ω̄ ε3^{m}ω ε3^{m}ω̄

It is then easy to see that there is a generalized perfect isometry between B_{0} and b_{0}: listing the irreducible characters of B_{0} and b_{0} in the same order as in the above tables, and relabelling them so that Irr(B_{0}) = {Ξ_{1}, …, Ξ_{q+7}} and Irr(b_{0}) = {X_{1}, …, X_{q+7}}, then the following is a generalized perfect isometry between B_{0} and b_{0}:

I: Ξ_{i} ⟼ ε_{i}X_{i} , i = 1, …, q + 7,

where ε_{i} = −1 if X_{i} ∈ \{α_{(q+3)/2}, …, α_{q−1}\}, ε_{i} = −ε if X_{i} ∈ \{μ_{1}, μ_{2}, \overline{μ_{1}}, \overline{μ_{2}}, χ, \overline{χ}\}, and ε_{i} = 1 otherwise.

We summarize the results we obtained in the following

Theorem 2.6. Suppose G = SU(3, q^{2}), Sz(q) or Re(q), and p is the defining characteristic of G. Let U be a Sylow p-subgroup of G, and B = N_{G}(U). Let B_{0} and b_{0} be the principal p-blocks of G and B respectively. Then there is a generalized perfect isometry with respect to p^{2}-regular elements between B_{0} and b_{0}.

Cartan Group, Generalized Characters

3.1 Cartan Group, Factors

In this section, we introduce the notions of Cartan group and Cartan matrix, as presented in [18].

Take G a finite group. Take C a closed set of conjugacy classes of G, and C′ = G \ C. Let Γ(C, G) = ((< χ, ψ >_{C}))_{χ,ψ∈Irr(G)} be the C-contribution matrix of G. Write ch(G) = ZIrr(G) the set of generalized characters of G. We define two Z-submodules of the space of complex class-functions of G: let

R(C) = {α^{C}, α ∈ ch(G)} =< χ^{C}, χ ∈ Irr(G) >_{Z}

and

P(C) = {β ∈ ch(G)| β ≡ 0 outside C}

(then P(C) is the Z-submodule of R(C) consisting of generalized characters).

The fact that C is closed implies (via Galois Theory) that the modules R(C) and P(C) have the same Z-rank, and that this rank is the number of conjugacy classes in C (cf [24]): we have

s := k(C) = rk_{Z}(Γ(C, G)) = rk_{Z}(R(C)) = rk_{Z}(P(C)).

The quotient R(C)/P(C) is thus an Abelian group, which we call the Cartan group of G (with respect to C), and denote by Cart(C, G), or just Cart(C). We have Cart(C) ≅ C_{d_{1}} × ⋯ × C_{d_{s}} (a product of s cyclic groups).

Now R(C), P(C) and Cart(C) have decompositions into direct sums corresponding to the C-blocks of G. For B a union of C-blocks of G, we denote by R(C, B), P(C, B) and Cart(C, B) the corresponding direct summands of R(C), P(C) and Cart(C) respectively.

For a given Z-basis {φ_{i}, 1 ≤ i ≤ r} of P(C, B), we define the Cartan matrix of B to be the matrix C(B) = ((< φ_{i}, φ_{j} >_{C}))_{1≤i,j≤r}. A different choice of Z-basis leads to a Cartan matrix which is equivalent over Z to C(B) (more precisely, if C′(B) is the new Cartan matrix, then there exists a unimodular integral matrix A such that C′(B) = A^{t}C(B)A). In particular, both Cartan matrices have the same invariant factors. The invariant factors are precisely the orders of the cyclic factors of Cart(C, B).

Now the invariant factors of C(B) are also linked to the invariant factors of the contribution matrix. There exists a Z-basis {ψ_{i}, 1 ≤ i ≤ r} for R(C, B) such that {d_{i}ψ_{i}, 1 ≤ i ≤ r} is a Z-basis for P(C, B). Then d_{r} is the smallest positive integer such that d_{r}Γ(C, B) has integer entries, and the non-zero invariant factors of d_{s}Γ(C, G) are 1 = \frac{d_{s}}{d_{r}} , \frac{d_{r}}{d_{r−1}} , …, \frac{d_{r}}{d_{1}} (cf [18], Lemma 1.3).

All these definitions and properties also apply to C′. From the definitions for P and R and the definition of blocks, we see that P(C, B) ⊕ P(C′, B) ⊂ R(G, B). Furthermore, using the fact that C is closed, Galois Theory implies that |G|R(G, B) ⊂ P(C, B) ⊕ P(C′, B). Hence we get that rk_{Z}(P(C, B)) + rk_{Z}(P(C′, B) = rk_{Z}(R(G, B)) = |B|. Thus, if Γ(C, B) has rank r, then Γ(C, B) has rank |B| − r.

Finally, note that, if C is the set of p-regular elements of G for some prime p, then the invariant factors of the Cartan matrix are known to be the orders of the p-defect groups of p-regular classes (cf e.g. Isaacs [16]).

3.2 Generalized Characters

3.2.1 Order in the Cartan group

We have seen that the exponent d_{s} of the Cartan group Cart(C) is the smallest positive integer such that d_{s}Γ(C, G) has integral entries. This implies that, for any χ ∈ Irr(G), the class function d_{s}χ^{C} is a generalized character. Indeed, if χ ∈ Irr(G), then χ^{C} is a class function of G, and we can write

χ^{C} = ∑_{ψ∈Irr(G)} < χ^{C}, ψ >_{G} ψ

= ∑_{ψ∈Irr(G)} < χ^{C}, ψ^{C} >_{G} ψ

= ∑_{ψ∈Irr(G)} < χ, ψ >_{C} ψ.

And, since d_{s} < χ, ψ >_{C}∈ Z for all ψ ∈ Irr(G), we see that d_{s}χ^{C} is a Z-linear combination of irreducible characters of G, i.e. a generalized character.

Given any χ ∈ Irr(G), we would like to find the smallest positive integer d such that dχ^{C} is a generalized character. The integer d will be called the order of χ in Cart(C).

From what we wrote above, we deduce that, for χ ∈ Irr(G) and d ∈ N,

dχ^{C} ∈ ch(G) ⟺ d < χ^{C}, ψ^{C} >∈ Z, ∀ψ ∈ Irr(G)
⟺ \{ \begin{matrix}d < χ^{C}, ψ^{C} >∈ Z, ∀ψ ∈ Irr(G) such \\ that ψ is directly C-linked to χ \end{matrix}

Thus the order of χ in Cart(C) is the smallest positive integer d such that d times the column of Γ(C, G) corresponding to χ has integral entries. Then, the decomposition of dχ^{C} as a linear combination of irreducible characters of G can be read from the contribution matrix. Only characters which are directly linked to χ will appear with non-zero coefficients. Finally, notice that, for χ ∈ Irr(G), since χ = χ^{C} + χ^{C′}, then χ has the same order in Cart(C) and Cart(C′).

3.2.2 First observations

Order of the trivial character

First note that, for each χ ∈ Irr(G), we can write χ^{C} = χ ⊗ 1_{G}^{C}, so that, for any d ∈ N, we have dχ^{C} = χ ⊗ (d1_{G}^{C}). Hence, if d1_{G}^{C} ∈ ch(G), then dχ^{C} ∈ ch(G) for all χ ∈ Irr(G). This implies that the trivial character 1_{G} has maximal order in Cart(C), and thus that the order of 1_{G} in Cart(C) is the exponent d_{s}, and that, for each χ ∈ Irr(G), the order of χ in Cart(C) divides d_{s}.

Characters of order 1

We next study C-blocks of G consisting of a single character. Suppose {χ} is such a C-block of G. Then < χ, ψ >_{C}= 0 for all ψ ∈ Irr(G), ψ ≠ χ. Hence we have χ^{C} = tχ for some t ∈ C. If we suppose furthermore that 1 ∈ C, then χ(1) = χ^{C}(1) = tχ(1) leads to t = 1 and χ^{C} = χ. Hence χ vanishes outside C, and χ has order 1 in Cart(C).

Conversely, if χ vanishes outside C, then χ^{C} = χ, so that χ has order 1 in Cart(C), and < χ, ψ >_{C}=< χ^{C}, ψ >_{G}=< χ, ψ >_{G}= 0 for all ψ ∈ Irr(G)\{χ}, so that {χ} is a C-block of G.

Finally, note that, if {χ} is a C-block of G, since < χ, ψ >_{G}=< χ, ψ >_{C} + < χ, ψ >_{C′} for all ψ ∈ Irr(G), we obtain that < χ, ψ >_{C′}= 0 for all ψ ∈ Irr(G) \ {χ}, so that {χ} is also a C′-block of G. Hence, if 1 ∉ C, then 1 ∈ C′ and χ^{C′} = χ.

Hence, for χ ∈ Irr(G), {χ} is a C-block of G if and only if χ vanishes outside C (if 1 ∈ C) or χ vanishes outside C′ (if 1 ∈ C′).

Now suppose that χ ∈ Irr(G) has order 1 in Cart(C). Then χ^{C} ∈ ch(G) and, in particular, < χ^{C}, χ >_{G}∈ Z. However, < χ^{C}, χ >_{G}= \frac{1}{|G|} ∑_{g∈C} |χ(g)|^{2}, so that 0 ≤< χ^{C}, χ >_{G}≤ 1. This yields that < χ^{C}, χ >_{G}∈ {0, 1}. Supposing 1 ∈ C, we have < χ^{C}, χ >_{G}≠ 0, so that < χ^{C}, χ >_{G}= 1. But then < χ^{C′}, χ >_{G}=< χ, χ >_{G} − < χ^{C}, χ >_{G}= 0 and, since < χ^{C′}, χ >_{G}= \frac{1}{|G|} ∑_{g∈C′} |χ(g)|^{2}, we obtain that χ vanishes identically on C′. Supposing 1 ∈ C′, we would obtain that χ vanishes identically on C.

Finally, we obtain

Proposition 3.1. Suppose 1 ∈ C. Then, for χ ∈ Irr(G), the following are equivalent:

• (i) {χ} is a C-block of G.

• (ii) χ vanishes outside C.

• (iii) χ has order 1 in Cart(C).

Blockwise considerations

We have seen that at least one character (namely the trivial one) must have maximal order d_{s}, which is the biggest invariant factor of the cartan matrix. However, it is not clear if, for example, for each invariant factor d_{i} of the Cartan matrix, there is an irreducible character whose order is d_{i} (and, in fact, it is false in general). However, by looking at things block by block, we can obtain a bit more information. Namely, if C is the set of p^{k}-regular elements, where p is a prime and k is a positive integer, then, in each C-block of G, there is (at least) one character whose order in Cart(C) is an invariant factor of the Cartan matrix.

Let B_{1}, …, B_{l} be the C-blocks of G. Then Γ(C, G) has a block-structure corresponding to the blocks of G:

Γ(C, G) = ( \begin{matrix}Γ_{1}(C) & & & & (0) \\ & . & . & . \\ (0) & & & & Γ_{l}(C) \end{matrix} )

where we write Γ_{i}(C) for the C-contribution matrix of B_{i}.

We have a similar decomposition for the Cartan group:

Cart(C, G) = ⨁_{i=1}^{l} Cart(C, B_{i}).

We let M(C) = d_{s}Γ(C, G) ∈ M_{n}(Z). The Smith Normal Form of M(C) is

D = ( 1 ; \frac{d_{s}}{d_{s−1}}  (0) ; ... ; \frac{d_{s}}{d_{1}} ; 0 ; (0)  ... ; 0 )

i.e. there exists U, V ∈ M_{n}(Z) unimodular integral matrices such that M(C) = d_{s}Γ(C, G) = UDV. And, in fact, because of the block-structure of M(C), there exist U_{1}, …, U_{l} and V_{1}, …, V_{l} unimodular integral matrices such that

M(C) = ( \begin{matrix}U_{1} & & & & (0) \\ & . & . & . \\ (0) & & & & U_{l} \end{matrix} ) ( \begin{matrix}D_{1} & & & & (0) \\ & . & . & . \\ (0) & & & & D_{l} \end{matrix} ) ( \begin{matrix}V_{1} & & & & (0) \\ & . & . & . \\ (0) & & & & V_{l} \end{matrix} )

where D_{i} is the Smith Normal Form of Γ_{i}(C). (Note that, in general,

V ≠ ( \begin{matrix}V_{1} & & & & (0) \\ & . & . & . \\ (0) & & & & V_{l} \end{matrix} ) (and similarly for U), since the columns and rows

may need to be reordered to obtain D from ( \begin{matrix}D_{1} & & & & (0) \\ & . & . & . \\ (0) & & & & D_{l} \end{matrix} ).)

Hence, for i ∈ {1, …, l}, we have

M_{i}(C) := d_{s}Γ_{i}(C) = U_{i}D_{i}V_{i} (†)

and

D_{i} = ( \frac{d_{s}}{d_{i,1}} ; ... ; \frac{d_{s}}{d_{i,k_{i}}} ; 0 ; ... ; 0 )

where

k_{i} = rk_{Z}(Γ_{i}(C)) (and thus s = ∑_{i=1}^{l} k_{i}),

{d_{i,1}, …, d_{i,k_{i}}} ⊂ {d_{1}, …, d_{s}} and d_{i,k_{i}}|d_{i,k_{i−1}}| ⋯ |d_{i,1} for all 1 ≤ i ≤ l,

and {d_{1}, …, d_{s}} = ⋃_{i=1}^{l}{d_{i,1}, …, d_{i,k_{i}}}.

Multiplying both sides of (†) by \frac{1}{d_{s}} Id, we obtain

Γ_{i}(C) = U_{i} ( \frac{1}{d_{i,1}} ; ... ; \frac{1}{d_{i,k_{i}}} ; 0 ; ... ; 0 ) V_{i}, for all 1 ≤ i ≤ l.

And, as U_{i} and V_{i} are invertible in some M_{n_{i}}(Z), we have

( \frac{1}{d_{i,1}} ; ... ; \frac{1}{d_{i,k_{i}}} ; 0 ; ... ; 0 ) = U_{i}^{−1}Γ_{i}(C)V_{i}.

Hence, for d ∈ N, dΓ_{i}(C) is integral if and only if \frac{d}{d_{i,1}} , …, \frac{d}{d_{i,k_{i}}} ∈ Z, and thus if and only if d_{i,1}|d. Hence d_{i,1} is the smallest positive integer such that d_{i,1}Γ_{i}(C) is integral.

In particular, if C is the set of p^{k}-regular elements of G, where p is a prime and k is a positive integer, then, since the order of any irreducible character of G in Cart(C) is a power of p (cf 3.2.3), we see that at least one character in B_{i} has order d_{i,1} in Cart(C) (the others having order dividing d_{i,1}).

3.2.3 The prime case

Let G be a finite group, and p be a prime number. We have the following classical result:

Theorem 3.2. If C is the set of p-regular elements of G and χ ∈ Irr(G), then the order of χ in the Cartan group Cart(C) is p^{d(χ)}, where d(χ) is the p-defect of χ (i.e. p^{d(χ)} = \frac{|G|_{p}}{χ(1)_{p}}).

A proof of this can be found for example in [20]. The first step is to show that |G|_{p}χ^{C} is a generalized character, and is just an application of Brauer’s Characterization of Characters (cf [20], Lemma (2.15)). Thus the order of χ in Cart(C) is a power of p. It follows from this and the properties of the discrete valuation ring used in the construction of the p-blocks of G that p^{d(χ)}χ^{C} is a generalized character, while p^{d(χ)−1}χ^{C} is not (cf [20], Lemma (3.23)).

We now turn to the situation where C is the set of p^{k}-regular elements of G, for some k ≥ 1. Then C is closed, and is a union of p-sections. This implies that |G|_{p}1_{C} is a generalized character of G (where 1_{C} is the characteristic function of C). Now, if χ ∈ Irr(G), then |G|_{p}χ^{C} = χ(|G|_{p}1_{C}) is also a generalized character.

On the other hand, for any ψ ∈ Irr(G), we have < χ^{C}, ψ >_{G}∈ Q, so that \frac{|G|}{χ(1)} < χ^{C}, ψ >_{G}∈ Q. But

\frac{|G|}{χ(1)} < χ^{C}, ψ >_{G}= ∑_{i=1}^{s}[G : C_{G}(y_{i})]\frac{χ(y_{i})}{χ(1)}\overline{ψ(y_{i})}

(where y_{1}, …, y_{s} are representatives for the conjugacy classes in C), so that \frac{|G|}{χ(1)} < χ^{C}, ψ >_{G} is an algebraic integer.

Hence \frac{|G|}{χ(1)} < χ^{C}, ψ >∈ Z for all ψ ∈ Irr(G), and thus \frac{|G|}{χ(1)}χ^{C} is a generalized character.

Hence |G|_{p}χ^{C} and \frac{|G|}{χ(1)}χ^{C} are generalized characters, and, since \frac{|G|_{p}}{χ(1)_{p}} =

gcd(|G|_{p}, \frac{|G|}{χ(1)}), we see that \frac{|G|_{p}}{χ(1)_{p}} χ^{C} = p^{d(χ)}χ^{C} is a generalized character. Thus the order of χ in Cart(C) divides p^{d(χ)}.

3.3 First Examples

In this section, we obtain information about the p^{k}-Cartan group and the orders of irreducible characters in some easy examples. The first is the case of Abelian groups; then we compute these informations for the three families of groups we studied in the previous part; finally, we study the particular case when the contribution matrix has Z-rank one.

3.3.1 Abelian groups

Let G be an Abelian finite group, and p be a prime number such that p^{k}||G| for some integer k ≥ 1. We take C to be the set of p^{k}-regular elements of G.

Then C is in fact a (normal) subgroup of G. Indeed, write G = G_{p}G_{p′}, and take g, h ∈ C; we can (uniquely) write g = g_{p}g_{p′} = g_{p′}g_{p} and h = h_{p}h_{p′} = h_{p′}h_{p}, with, as g and h are p^{k}-regular, o(g_{p}) < p^{k} and o(h_{p}) < p^{k}. Then gh = (g_{p}h_{p})(g_{p′}h_{p′}) = (g_{p′}h_{p′})(g_{p}h_{p}), with g_{p}h_{p} ∈ G_{p}, g_{p′}h_{p′} ∈ G_{p′}, and

(o(gh))_{p} = o(g_{p}h_{p}) ≤ Max(o(g_{p}), o(h_{p})) ≤ p^{k}

(the first inequality being true since both o(g_{p}) and o(h_{p}) are powers of p, so that lcm(o(g_{p}), o(h_{p})) = Max(o(g_{p}), o(h_{p}))). Of course, g^{−1} ∈ C and 1 ∈ C.

Hence C is a normal subgroup of G, and, furthermore, G = C_{G}(C). Thus, by Clifford’s Theory, we see that, for all λ ∈Irr(C), Ind_{C}^{G}(λ) has [G : C] irreducible summands, each appearing with multiplicity 1 (since G is Abelian and all its irreducible characters have degree 1). And, if χ ∈ Irr(G) is such an irreducible summand, then Res_{C}^{G}(χ) = λ.

Furthermore, each irreducible character of G appears in this way. We can label each irreducible character of G with an irreducible character of C and an integer i, 1 ≤ i ≤ [G : C]. We write

Irr(G) = {χ_{λ}^{(i)}| λ ∈ Irr(C), 1 ≤ i ≤ [G : C]},

where, for each λ ∈ Irr(C), Ind_{C}^{G}(λ) = ∑_{i=1}^{[G:C]} χ_{λ}^{(i)}.

Now, for all χ_{λ}^{(i)}, χ_{μ}^{(j)} ∈ Irr(G),

< χ_{λ}^{(i)}, χ_{μ}^{(j)} >_{C} = \frac{1}{|G|} ∑_{g∈C} χ_{λ}^{(i)}(g)χ_{μ}^{(j)}(g^{−1})

= \frac{1}{|G|} ∑_{g∈C} Res_{C}^{G}(χ_{λ}^{(i)})(g)Res_{C}^{G}(χ_{μ}^{(j)})(g^{−1})

= \frac{1}{|G|} ∑_{g∈C} λ(g)μ(g^{−1}).

Hence

< χ_{λ}^{(i)}, χ_{μ}^{(j)} >_{C}= \frac{1}{[G : C]} < λ, μ >_{C} . (†)

We see that χ_{λ}^{(i)} and χ_{μ}^{(j)} belong to the same C-block of G if and only if λ = μ. Hence G has |C| C-blocks, which we write {B_{λ}, λ ∈ Irr(C)}, each containing [G : C] irreducible characters:

B_{λ} = {χ_{λ}^{(i)}, 1 ≤ i ≤ [G : C]}, ∀λ ∈ Irr(C).

It is therefore clear that all irreducible characters of G in a given C-block have the same order in the Cartan group Cart(C, G) (since they have the same restriction to C). It is also clear from (†) that this order is [G : C].

Note that, from (†), we deduce that all entries in the contribution matrix of any given C-block of G will be the same, namely \frac{1}{[G:C]} (in particular, the contribution matrix of each C-block has rank 1). This in turn implies that the elementary divisors of the C-Cartan matrix (there are |C| of them) will all be equal to [G : C], and that the Cartan group Cart(C, G) is (isomorphic to) the direct product of |C| copies of the cyclic group of order [G : C].

Finally, note that, writing |G| = p^{a}m, with gcd(p, m) = 1, we have

|C| = mp^{(k−1)( \begin{matrix}a \\ a − k + 1 \end{matrix} )} = mp^{(k−1)( \begin{matrix}a \\ k − 1 \end{matrix} )}

and

[G : C] = p^{a−(k−1)( \begin{matrix}a \\ k − 1 \end{matrix} )}.

3.3.2 Special unitary groups, Suzuki groups, Ree groups

Using the results of the previous part, it is easy to compute the C-contributions in the case where G is SU(3, q^{2}), Sz(q) or Re(q), p is the defining characteristic, and C is the set of p^{2}-singular or p^{2}-regular elements of G. The orders of the irreducible characters of the principal p-block B_{0} of G in Cart(C, B_{0}) can be read directly from the contribution matrix, and the structure of the Cartan group can be obtained by reducing a multiple of the contribution matrix to Smith normal form (cf 3.1.1). The reduction to Smith normal form is performed by elementary row and column operation.

We will use the notations of the previous part: in each case, we write U a Sylow p-subgroup of G and B = N_{G}(U); we let B_{0} and b_{0} be the principal p-blocks of G and B respectively; we let C and D be the sets of p^{2}-singular elements of G and B respectively, and we let C′ = G \ C and D′ = B \ D.

In the three cases, we prefer to compute the Smith normal form of the matrix associated to the D-contributions of Irr(b_{0}), as this is easier to manipulate than for B_{0}. Because of the perfect isometry we exhibited, this doesn’t change anything to the results we’re looking for.

The case of SU(3, q^{2})

In this section, we let G = SU(3, q^{2}) and d = gcd(3, q + 1). With the values taken by the irreducible characters of b_{0} on the elements of order divisible by 82 PART 3. CARTAN GROUP, GENERALIZED CHARACTERS 4, we compute the contribution matrix Γ(D, b_{0}). For any χ, ψ ∈ b_{0}, we have

< χ, ψ >_{D}= \frac{1}{|B|} ∑_{y∈D} χ(y)\overline{ψ(y)}.

If d = 1, then D is only one conjugacy class (namely B_{3}^{(0,0)}), so that

< χ, ψ >_{D} = \frac{|D|}{|B|}χ(D)\overline{ψ(D)}
= \frac{q(q^{2}−1)}{q^{3}(q^{2}−1)}χ(D)ψ(D)
= \frac{1}{q^{2}} χ(B_{3}^{(0,0)})ψ(B_{3}^{(0,0)}).

If d = 3, then we have, for any χ, ψ ∈ b_{0},

< χ, ψ >_{D} = \frac{1}{|B|} ∑_{y∈D} χ(y)\overline{ψ(y)}
= \frac{1}{|B|} ∑_{0≤k,l≤2} |B_{3}^{(k,l)}|χ(B_{3}^{(k,l)})ψ(B_{3}^{(k,l)})
= \frac{1}{3q^{2}} ∑_{0≤k,l≤2} χ(B_{3}^{(k,l)})ψ(B_{3}^{(k,l)}).

In particular, for any 0 ≤ v ≠ v′ ≤ 2,

< θ_{(q^{2}−1)/3}^{(0,v)}, θ_{1}^{(u)} >_{D}= \frac{1}{3q^{2}} (3(q − \frac{q + 1}{3}) + 6(−\frac{q + 1}{3})) = \frac{1}{3q^{2}} (−3)

< θ_{(q^{2}−1)/3}^{(0,v)}, θ_{(q^{2}−1)/3}^{(0,v)} >_{D}= \frac{1}{3q^{2}} (3(q − \frac{q + 1}{3})^{2} + 6(−\frac{q + 1}{3})^{2}) = \frac{1}{3q^{2}} (2q^{2} + 1)

< θ_{(q^{2}−1)/3}^{(0,v)}, θ_{(q^{2}−1)/3}^{(0,v′)} >_{D}= \frac{1}{3q^{2}} (6(−\frac{q + 1}{3})(q − \frac{q + 1}{3}) + 3(−\frac{q + 1}{3})^{2}) = \frac{1}{3q^{2}} (1−q^{2})

We obtain, labelling the lines and columns in the order we used to list the irreducible characters of b_{0} in the previous part: if d = 1,

Γ(D, b_{0}) = \frac{1}{q^{2}} ( \begin{matrix}0 & ⋯ & 0 & 0 & ⋯ & 0 & 0 \\ ⋮ & & ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & ⋯ & 0 & 0 & ⋯ & 0 & 0 \\ 0 & ⋯ & 0 & 1 & ⋯ & 1 & −1 \\ ⋮ & & ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & ⋯ & 0 & 1 & ⋯ & 1 & −1 \\ 0 & ⋯ & 0 & −1 & ⋯ & −1 & 1 \end{matrix} )

q + 1  q^{2} − 1  1

and if d = 3,

Γ(D, b_{0}) = \frac{1}{3q^{2}} ( \begin{matrix}0 & ⋯ & 0 & 0 & ⋯ & 0 & 0 & 0 & 0 \\ ⋮ & & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & ⋯ & 0 & 0 & ⋯ & 0 & 0 & 0 & 0 \\ 0 & ⋯ & 0 & 9 & ⋯ & 9 & −3 & −3 & −3 \\ ⋮ & & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & ⋯ & 0 & 9 & ⋯ & 9 & −3 & −3 & −3 \\ 0 & ⋯ & 0 & −3 & ⋯ & −3 & 2q^{2} + 1 & 1 − q^{2} & 1 − q^{2} \\ 0 & ⋯ & 0 & −3 & ⋯ & −3 & 1 − q^{2} & 2q^{2} + 1 & 1 − q^{2} \\ 0 & ⋯ & 0 & −3 & ⋯ & −3 & 1 − q^{2} & 1 − q^{2} & 2q^{2} + 1 \end{matrix} )

\frac{q+1}{3}  ^{\frac{q^{2}−1}{3}}  3

In the array below, we show, for each C-block of B_{0}, the 2-defect of the characters belonging to the block and their order in the Cartan group (these turn out to be the same for each character in a given C-block). Recall that, for each character χ ∈ b_{0}, the order of χ in Cart(C, B_{0}) (which is also its order in Cart(C′, B_{0})) is the smallest positive integer δ such that δ times the column of Γ(C, B_{0}) is integral. Because of the generalized perfect isometry between B_{0} and b_{0}, the corresponding character of b_{0} has the same order in Cart(D, b_{0}). It also turns out that this doesn’t depend on the value of d.

C-block  p^{d_{2}(χ)} order in Cart(C, B_{0})
{χ_{q^{2}−q}}; {χ_{q(q^{2}−q+1)}^{(u)}}, 1 ≤ u ≤ q q^{2}  1
{All the rest of B_{0}}  q^{3}  q^{2}

We want now to compute the invariant factors {d_{1}, …, d_{s}} of the Cartan matrix C_{0}(D) of b_{0}. The biggest one, d_{s}, is the smallest positive integer such that d_{s}Γ(D, b_{0}) = M(D) is an integral matrix.

It is easy to see that, whatever the value of d is, d_{s} = q^{2} (just notice that, if d = 3, then q ≡ −1 (mod 3), so that 2q^{2} + 1 and 1 − q^{2} are divisible by 3). The non-zero invariant factors of M(D) are \{1, \frac{d_{s}}{d_{s−1}} , … \frac{d_{s}}{d_{1}} \}.

If d = 1, the last column (resp. line) of M(D) is the opposite of any other non-zero column (resp. line) of M(D). Hence there exists an integral matrix V such that

M(D) = V^{t} ( \begin{matrix}1 \\ & 0 \\ & & . & . & . \\ & & & & & 0 \end{matrix} ) V.

Thus C_{0}(D) has exactly one non-zero invariant factor, d_{s} = q^{2}. We have Cart(D, b_{0}) ≅ Cart(C, B_{0}) ≅ C_{q^{2}}.

Suppose now that d = 3. We will denote by C_{i} and R_{i} the i-th column and row of any matrix. Performing C_{i} ⟵ C_{i} − C_{(q+q^{2})/3} and R_{i} ⟵ R_{i} − R_{(q+q^{2})/3} for \frac{q+1}{3} + 1 ≤ i ≤ \frac{q+q^{2}}{3} − 1, M(D) becomes

( \begin{matrix}(0) & (0) \\ (0) & \begin{matrix}3 & −1 & −1 & −1 \\ −1 & \frac{2q^{2}+1}{3} & \frac{1−q^{2}}{3} & \frac{1−q^{2}}{3} \\ −1 & \frac{1−q^{2}}{3} & \frac{2q^{2}+1}{3} & \frac{1−q^{2}}{3} \\ −1 & \frac{1−q^{2}}{3} & \frac{1−q^{2}}{3} & \frac{2q^{2}+1}{3} \end{matrix} \end{matrix} )

Then, R_{(q+q^{2})/3} ⟵ R_{(q+q^{2})/3}+∑(non-zero R_{i}’s) and C_{(q+q^{2})/3} ⟵ C_{(q+q^{2})/3}+ ∑(non-zero C_{i}’s) give

( \begin{matrix}(0) & (0) \\ (0) & \begin{matrix}\frac{2q^{2}+1}{3} & \frac{1−q^{2}}{3} & \frac{1−q^{2}}{3} \\ \frac{1−q^{2}}{3} & \frac{2q^{2}+1}{3} & \frac{1−q^{2}}{3} \\ \frac{1−q^{2}}{3} & \frac{1−q^{2}}{3} & \frac{2q^{2}+1}{3} \end{matrix} \end{matrix} )

We then work on the only lower right block. C_{1} ⟵ C_{1} + 2C_{2} gives

( \begin{matrix}1 & \frac{1−q^{2}}{3} & \frac{1−q^{2}}{3} \\ q^{2} + 1 & \frac{2q^{2}+1}{3} & \frac{1−q^{2}}{3} \\ 1 − q^{2} & \frac{1−q^{2}}{3} & \frac{2q^{2}+1}{3} \end{matrix} ) \underrightarrow{C_{2,3} ← C_{2,3} − \frac{1 − q^{2}}{3} C_{1}} ( \begin{matrix}1 & 0 & 0 \\ q^{2} + 1 & \frac{q^{2}(q^{2}+2)}{3} & \frac{q^{2}(q^{2}−1)}{3} \\ 1 − q^{2} & \frac{q^{2}(1−q^{2})}{3} & \frac{q^{2}(4−q^{2})}{3} \end{matrix} )

Performing R_{2} ⟵ R_{2} − (q^{2} + 1)R_{1} and R_{3} ⟵ R_{3} − (1 − q^{2})R_{1} produces

( \begin{matrix}1 & 0 & 0 \\ 0 & \frac{q^{2}(q^{2}+2)}{3} & \frac{q^{2}(q^{2}−1)}{3} \\ 0 & \frac{q^{2}(1−q^{2})}{3} & \frac{q^{2}(4−q^{2})}{3} \end{matrix} ) \underrightarrow{C_{2} ← C_{2} − C_{3}} ( \begin{matrix}1 & 0 & 0 \\ 0 & q^{2} & \frac{q^{2}(q^{2}−1)}{3} \\ 0 & −q^{2} & \frac{q^{2}(4−q^{2})}{3} \end{matrix} )

Finally, R_{3} ⟵ R_{3} + R_{2} and C_{3} ⟵ C_{3} − \frac{q^{2}−1}{3}C_{2} give

( \begin{matrix}1 & 0 & 0 \\ 0 & q^{2} & 0 \\ 0 & 0 & q^{2} \end{matrix} )

Whence we find that s = rk_{Z}(Γ(D, b_{0}) = 3, and the non-zero invariant factors of the Cartan matrix C_{0}(D) are d_{3} = q^{2} and d_{2} = d_{1} = 1. Here again, we have Cart(D, b_{0}) ≅ Cart(C, B_{0}) ≅ C_{q^{2}}.

We now turn to elements of order not divisible by 4. We have seen from the definition that b_{0} and B_{0} are also perfectly isometric with respect to C′ and D′. Hence the Cartan groups Cart(C′, B_{0}) and Cart(D′, b_{0}) are isomorphic. We have Γ(D′, b_{0}) = Id − Γ(D, b_{0}), and we shall find the |Irr(b_{0})| − d = \frac{q^{2}+q}{d} non-zero invariant factors of q^{2}Γ(D′, b_{0}) = N(D′).

If d = 1, then

N(D′) = ( \begin{matrix}q^{2} & & (0) & & & & 0 \\ & ... & & & (0) & & ⋮ \\ (0) & & q^{2} & & & & 0 \\ & & & q^{2} − 1 & & (−1) & 1 \\ & (0) & & & ... & & ⋮ \\ & & & (−1) & & q^{2} − 1 & 1 \\ 0 & ⋯ & 0 & 1 & ⋯ & 1 & q^{2} − 1 \end{matrix} )

We reduce the lower-right (q^{2} by q^{2}) block to Smith normal form. Reversing the order of the columns, we get

( \begin{matrix}1 & & & & q^{2} − 1 \\ ⋮ & (−1) & & q^{2} − 1 \\ ⋮ & & ↗ \\ 1 & q^{2} − 1 & & (−1) \\ q^{2} − 1 & 1 & ⋯ & ⋯ & 1 \end{matrix} )

Then, C_{i} ⟵ C_{i} + C_{1}, 1 < i ≤ q^{2} gives

( \begin{matrix}1 & & & & q^{2} \\ ⋮ & (0) & & q^{2} \\ ⋮ & & ↗ \\ 1 & q^{2} & & (0) \\ q^{2} − 1 & q^{2} & ⋯ & ⋯ & q^{2} \end{matrix} )

R_{q^{2}} ⟵ R_{q^{2}} − ∑_{i=1}^{q^{2}−1} R_{i} gives

( \begin{matrix}1 & & & & q^{2} \\ ⋮ & (0) & & q^{2} \\ ⋮ & & ↗ \\ 1 & q^{2} & & (0) \\ 0 & 0 & ⋯ & ⋯ & 0 \end{matrix} )

R_{i} ⟵ R_{i} − R_{1}, 1 < i < q^{2}, gives

( \begin{matrix}1 & 0 & ⋯ & 0 & q^{2} \\ 0 & (0) & & q^{2} & −q^{2} \\ ⋮ & & ↗ \\ ⋮ & q & & (0) & −q^{2} \\ 0 & ⋯ & ⋯ & ⋯ & 0 \end{matrix} )

C_{q^{2}} ⟵ C_{q^{2}} − q^{2}C_{1} gives

( \begin{matrix}1 & 0 & ⋯ & ⋯ & 0 \\ 0 & & & q^{2} & −q^{2} \\ ⋮ & & ↗ & & ⋮ \\ ⋮ & q^{2} & & & −q^{2} \\ 0 & ⋯ & ⋯ & ⋯ & 0 \end{matrix} )

C_{q^{2}} ⟵ C_{q^{2}} + ∑_{i=2}^{q^{2}−1} C_{i} gives

( \begin{matrix}1 & 0 & ⋯ & ⋯ & 0 \\ 0 & & & q^{2} & 0 \\ ⋮ & & ↗ & & ⋮ \\ ⋮ & q^{2} & & & 0 \\ 0 & ⋯ & ⋯ & ⋯ & 0 \end{matrix} )

which, reversing the order of the rows, apart from the first, gives finally

( \begin{matrix}1 \\ & q^{2} \\ & & . & . & . \\ & & & & & q^{2} \\ & & & & & & 0 \end{matrix} )

We obtain that N(D′) is equivalent (over Z) to

( \begin{matrix}q^{2} & & (0) & & & & & 0 \\ & ... & & & (0) & & & ⋮ \\ (0) & & q^{2} & & & & & 0 \\ & & & 1 & & (0) & & 0 \\ & (0) & & & q^{2} & & & ⋮ \\ & & & & & ... & & ⋮ \\ & & & (0) & & & q^{2} & 0 \\ 0 & ⋯ & 0 & 0 & ⋯ & ⋯ & 0 & 0 \end{matrix} )

so that the non-zero invariant factors of the Cartan matrix C_{0}(D′) of b_{0} are d′_{1} = … = d′_{q^{2}+q−1} = 1 and d′_{q^{2}+q} = q^{2}, and Cart(D′, b_{0}) ≅ Cart(C′, B_{0}) ≅ C_{q^{2}}.

If d = 3, then

q^{2}Γ(D′, b_{0}) = ( \begin{matrix}q^{2} \\ & ... & & & (0) & & (0) \\ & & q^{2} \\ & & & q^{2} − 3 & & (−3) \\ & (0) & & & ... & & (1) \\ & & & (−3) & & q^{2} − 3 \\ & (0) & & & (1) & & (\frac{q^{2}−1}{3}) \end{matrix} )

\frac{q+1}{3}  ^{\frac{q^{2}−1}{3}}  3

By working on the last 3 rows and columns (which are the same), this can be reduced to

( \begin{matrix}q^{2} \\ & ... & & (0) \\ & & q^{2} \\ & (0) & & \begin{aligned}q^{2} − 3 & & (−3) & 1 \\ & ... & & ⋮ & (0) \\ (−3) & & q^{2} − 3 & 1 \\ 1 & ⋯ & 1 & \frac{q^{2}−1}{3} & (0) \\ & (0) & & (0) & 0 0 0 0 \end{aligned} \end{matrix} )

Subtracting to the last non-zero column the sum of the preceeding \frac{q^{2}−1}{3} columns, and then doing the same thing to the last non-zero row, we obtain(leaving aside the upper left (diagonal) block)

( \begin{matrix}q^{2} − 3 & & (−3) \\ & ... & & (0) \\ (−3) & & q^{2} − 3 \\ & (0) & & (0) \end{matrix} )

Now we work separately on the non-zero block. C_{1} ⟵ C_{1} + ∑_{i>1} C_{i} gives

( \begin{matrix}1 & −3 & ⋯ & −3 \\ 1 & q^{2} − 3 & & (−3) \\ ⋮ & & ... \\ 1 & (−3) & & q^{2} − 3 \end{matrix} ) \underrightarrow{C_{i} ← C_{i} + 3C_{1}, i > 1} ( \begin{matrix}1 \\ 1 & q^{2} & & (0) \\ ⋮ & & ... \\ 1 & (0) & & q^{2} \end{matrix} )

And, finally, R_{i} ⟵ R_{i} − R_{1} for i > 1 gives

( \begin{matrix}1 \\ & q^{2} \\ & & . & . & . \\ & & & & & q^{2} \end{matrix} )

Putting all the blocks back together, we see that the non-zero invariant factors of the Cartan matrix C_{0}(D′) are d′_{1} = … = d′_{\frac{q+q^{2}}{3} −1} = 1 and d′_{\frac{q+q^{2}}{3}} = q^{2},

and, here again, Cart(D′, b_{0}) ≅ Cart(C′, B_{0}) ≅ C_{q^{2}}.

The case of Sz(q)

In this section, we let G = Sz(q), with q = 2^{2n+1}. With the values taken by the irreducible characters of b_{0} on the elements of order divisible by 4, we compute the contribution matrix Γ(D, b_{0}). We have, for any χ, ψ ∈ Irr(B),

< χ, ψ >_{D} = \frac{1}{|B|} ∑_{y∈D} χ(y)\overline{ψ(y)}
= \frac{1}{|B|} (\frac{q(q − 1)}{2}(χ(ρ)\overline{ψ(ρ)} + χ(ρ^{−1})\overline{ψ(ρ^{−1})}))
= \frac{1}{2q} (χ(ρ)ψ(ρ^{−1}) + χ(ρ^{−1})ψ(ρ))

We obtain, labelling the lines and columns in the order we used to list the irreducible characters of b_{0} in the previous part:

Γ(D, b_{0}) = \frac{1}{q} ( \begin{matrix}1 & −1 · · · · · · − 1 & 0 0 \\ −1 \\ ⋮ & (1)_{q−1,q−1} & (0) \\ −1 \\ 0 & 0 · · · · · · 0 & \frac{q}{2} \frac{−q}{2} \\ 0 & 0 · · · · · · 0 & \frac{−q}{2} \frac{q}{2} \end{matrix} ) .

In the array below, we show, for each C-block of B_{0}, the 2-defect of the characters belonging to the block and their order in the Cartan group. Here again, these turn out to be the same for each character in a given C-block.

C-block  p^{d_{2}(χ)} order in Cart(C, B_{0})
{1_{G}, Ω_{s}, Θ_{l}, Λ_{u}} q^{2}  q
{Γ_{1}, Γ_{2}}  q2^{n+1}  2

The only irreducible character of G which doesn’t belong to the principal 2-block is Π. It has 2-defect 0, so is a 2-block and a C-block by itself, and has order 1 in the Cartan group Cart(C, G).

We want now to compute the invariant factors {d_{1}, …, d_{s}} of the Cartan matrix C_{0}(D) of b_{0}. The biggest, d_{s}, is the smallest positive integer such that d_{s}Γ(D, b_{0}) = M(D) is an integral matrix. It is easy to see that, d_{s} = q. The non-zero invariant factors of M(D) are \{1, \frac{d_{s}}{d_{s−1}} , … \frac{d_{s}}{d_{1}} \}.

We will denote by C_{i} and R_{i} the i-th column and row of any matrix. We have

qΓ(D, b_{0}) = ( \begin{matrix}1 & −1 · · · · · · − 1 & 0 0 \\ −1 \\ ⋮ & (1)_{q−1,q−1} & (0) \\ −1 \\ 0 & 0 · · · · · · 0 & \frac{q}{2} \frac{−q}{2} \\ 0 & 0 · · · · · · 0 & \frac{−q}{2} \frac{q}{2} \end{matrix} )

Performing C_{i} ⟵ C_{i} + C_{1} for i > 1, we obtain

( \begin{matrix}1 & 0 · · · · · · 0 & 0 0 \\ −1 \\ ⋮ & (0)_{q−1,q−1} & (0) \\ −1 \\ 0 & 0 · · · · · · 0 & \frac{q}{2} \frac{−q}{2} \\ 0 & 0 · · · · · · 0 & \frac{−q}{2} \frac{q}{2} \end{matrix} )

Performing R_{i} ⟵ R_{i} + R_{1} for i > 1, we get

( \begin{matrix}1 & 0 · · · · · · 0 & 0 0 \\ 0 \\ ⋮ & (0)_{q−1,q−1} & (0) \\ 0 \\ 0 & 0 · · · · · · 0 & \frac{q}{2} \frac{−q}{2} \\ 0 & 0 · · · · · · 0 & \frac{−q}{2} \frac{q}{2} \end{matrix} )

Similarly, C_{q+2} ⟵ C_{q+2} + C_{q+1} and R_{q+2} ⟵ R_{q+2} + R_{q+1} lead to

( \begin{matrix}1 & 0 · · · · · · 0 & 0 0 \\ 0 \\ ⋮ & (0)_{q−1,q−1} & (0) \\ 0 \\ 0 & 0 · · · · · · 0 & \frac{q}{2} 0 \\ 0 & 0 · · · · · · 0 & 0 0 \end{matrix} ) .

Finally, there exists an integral matrix A such that

qΓ(D, b_{0}) = A^{t} ( 1 ; \frac{q}{2} ; 0 ; ... ; 0 ) A

whence s = 2, and the non-zero invariant factors of C_{0}(D) are d_{2} = q and d_{1} = 2. We have Cart(C, B_{0}) ≅ Cart(D, b_{0}) ≅ C_{q} × C_{2}.

We now turn to the sets C′ and D′ of elements of order not divisible by 4 in G and B respectively. We want to compute the (common) invariant factors {d′_{1}, …, d′_{s′}} of C_{0}(C′) and C_{0}(D′). From the equality

rk_{Z}(R(b_{0}, D)) + rk_{Z}(R(b_{0}, D′)) = |b_{0}| = |B_{0}| = q + 2

we see that s′ = q.

We have

qΓ(D′, b_{0}) = ( \begin{matrix}q − 1 & 1 & · · · · · · & 1 & 0 0 \\ 1 & q − 1 & & (−1) \\ ⋮ & & ... & & (0) \\ 1 & (−1) & & q − 1 \\ 0 & 0 & · · · · · · & 0 & \frac{q}{2} \frac{q}{2} \\ 0 & 0 & · · · · · · & 0 & \frac{q}{2} \frac{q}{2} \end{matrix} )

so that it is clear that the biggest invariant factor we will find is d′_{q} = q. We work separately on the two blocks of qΓ(D, b_{0}). The upper-left block has the same structure as the matrix N(D′) we obtained for SU(3, q^{2}) in the case d = 1. Reducing it to Smith normal form, we obtain

( \begin{matrix}1 \\ & q \\ & & . & . & . \\ & & & & & q \\ & & & & & & 0 \end{matrix} )

As for the second block, it is easy to see that, performing C_{2} ⟵ C_{2} − C_{1} and R_{2} ⟵ R_{2} − R_{1}, we obtain

( \begin{matrix}\frac{q}{2} & \frac{q}{2} \\ \frac{q}{2} & \frac{q}{2} \end{matrix} ) ⟼ ( \begin{matrix}\frac{q}{2} & 0 \\ 0 & 0 \end{matrix} )

Whence we get that qΓ(D′, b_{0}) is equivalent (over Z) to

( 1 ; \frac{q}{2} ; q ; ... ; q ; 0 ; 0 )

This gives us that the invariant factors of the Cartan matrices C_{0}(D′) and C_{0}(C′) are d′_{1} = ⋯ = d′_{q−2} = 1, d′_{q−1} = 2, d′_{q} = q, and Cart(C′, B_{0}) ≅ Cart(D′, b_{0}) ≅ C_{q} × C_{2}.

The case of Re(q)

In this section, we let G = Re(q), with q = 3^{2m+1}. Then D is the set of elements of order divisible by 9 in B. We first compute the contribution matrix Γ(D, b_{0}). For any γ, δ ∈ b_{0}, the D-contribution of γ and δ is given by

< γ, δ >_{D}= \frac{1}{|B|} ∑_{y∈D} γ(y)\overline{δ(y)} = \frac{1}{3q} ∑_{i=1}^{3} γ(x_{i})\overline{δ(x_{i})}.

In particular, < ψ, δ >_{D}= 0 for all δ ∈ b_{0}. Furthermore, if γ ∈ \{μ_{1,2}, \overline{μ_{1,2}}, χ, \overline{χ}\} and δ ∈ {ψ, α_{i}, λ}, then < γ, δ >_{D}= 0 (for we can factorize by 1+ω+ω̄ = 0). We then compute

< α_{i}, α_{j} >_{D}= \frac{1}{3q} 3 , < λ, λ >_{D}= \frac{1}{3q} 3 , < α_{i}, λ >_{D}= \frac{1}{3q} (−3) ,

< μ_{i}, μ_{j} >_{D}= \frac{1}{3q} 3^{2m}(1+1+1) = \frac{1}{3q} q , < μ_{i}, \overline{μ_{j}} >_{D}= \frac{1}{3q} 3^{2m}(1+ω̄^{2}+ω^{2}) = 0 ,

< χ, χ >_{D}= \frac{1}{3q} 3^{2m}3 = \frac{1}{3q} q , < χ, \overline{χ} >_{D}= 0 ,

< μ_{i}, χ >_{D}= \frac{1}{3q} (−3^{2m})(1+1+1) = \frac{1}{3q} (−q) , < μ_{i}, \overline{χ} >_{D}= \frac{1}{3q} (−3^{2m})(1+ω̄^{2}+ω^{2}) = 0.

The other contributions are found using these and complex conjugation. Finally, we obtain, labelling the lines and columns in the order used in the character table we gave in the previous part,

Γ(D, b_{0}) = \frac{1}{3q} ( \begin{matrix}0 & 0 ⋯ 0 & 0 & 0 0 & 0 0 & 0 & 0 \\ 0 & 3 ⋯ 3 & −3 & 0 0 & 0 0 & 0 & 0 \\ ⋮ & ⋮ ⋮ & ⋮ & ⋮ ⋮ & ⋮ ⋮ & ⋮ & ⋮ \\ 0 & 3 ⋯ 3 & −3 & 0 0 & 0 0 & 0 & 0 \\ 0 & −3 ⋯ − 3 & 3 & 0 0 & 0 0 & 0 & 0 \\ 0 & 0 ⋯ 0 & 0 & q q & 0 0 & −q & 0 \\ 0 & 0 ⋯ 0 & 0 & q q & 0 0 & −q & 0 \\ 0 & 0 ⋯ 0 & 0 & 0 0 & q q & 0 & −q \\ 0 & 0 ⋯ 0 & 0 & 0 0 & q q & 0 & −q \\ 0 & 0 ⋯ 0 & 0 & −q −q & 0 0 & q & 0 \\ 0 & 0 ⋯ 0 & 0 & 0 0 & −q −q & 0 & q \end{matrix} )

We can now read the C-blocks of B_{0}, and, for each given block, the 3defect of its characters and their order in the Cartan group Cart(C, B_{0}) (once again, defect and order are in fact the same for any character in the block):

C-block  p^{d_{3}(χ)} order in Cart(C, B_{0})
{ξ_{4}} q^{2} 1
{ξ_{5}, ξ_{6}, ξ_{9}}; {ξ_{7}, ξ_{8}, ξ_{10}} q^{2}3^{k+1}  3
{ξ_{1}, ξ_{2}, η_{r}, η′_{r}, η_{t}, η′_{t}, η_{i}^{−}, η_{i}^{+}} q^{3}  q

We now turn to the invariant factors of the Cartan matrix. We see that the smallest (positive) integer d_{s} such that M(D) = d_{s}Γ(D, b_{0}) is an integral matrix is d_{s} = q, and that s = rk(Γ(D, b_{0})) = 3 (it suffices to consider the columns corresponding to χ, \overline{χ} and λ, the others being opposite to these columns). It is then easy to reduce M(D) = qΓ(D, b_{0}) to Smith normal form; M(D) is equivalent over Z to

( \begin{matrix}1  (0) ; \frac{q}{3} ; (0)  \frac{q}{3} & (0) \\ (0) & (0) \end{matrix} )

so that \{\frac{d_{s}}{d_{s}} = 1, \frac{d_{s}}{d_{s−1}} , \frac{d_{s}}{d_{s−2}} \} = \{1, \frac{q}{3}, \frac{q}{3}\}, the invariant factors of the Cartan matrix C_{0}(D) are d_{1} = d_{2} = 3 and d_{3} = q, and Cart(C, B_{0}) ≅ Cart(D, b_{0}) ≅ C_{q} × C_{3} × C_{3}.

Let us now turn to elements whose order is not divisible by 9. We have Γ(D′, b_{0}) = Id − Γ(D, b_{0}), whence we see that d′_{s′} = q (where, this time, s′ = rk(Γ(D′, b_{0})) = q + 4), and, after having re-ordered lines and columns, we have, writing N(D′) = d′_{s′}Γ(D′, b_{0}),

N(D′) = ( q ; \begin{matrix}q − 1 & 1 & ⋯ & 1 \\ 1 & q − 1 & & (−1) \\ ⋮ & & ... \\ 1 & (−1) & & q − 1 \end{matrix} ; \begin{matrix}\frac{2q}{3} & \frac{q}{3} & \frac{q}{3} \\ \frac{q}{3} & \frac{2q}{3} & −\frac{q}{3} \\ \frac{q}{3} & −\frac{q}{3} & \frac{2q}{3} \end{matrix} ; \begin{matrix}\frac{2q}{3} & \frac{q}{3} & \frac{q}{3} \\ \frac{q}{3} & \frac{2q}{3} & −\frac{q}{3} \\ \frac{q}{3} & −\frac{q}{3} & \frac{2q}{3} \end{matrix} )

The second block on the diagonal is the same as the one we considered in the case of Suzuki groups, and has Smith normal form

\begin{aligned}1 \\ & q & & & & (0) \\ & & . & . & . \\ & (0) & & & & q \\ & & & & & & 0 \end{aligned}

It just remains to study the lower right block. We write C_{i} (resp. R_{i}) for the i-th column (resp. row) of any matrix.

\begin{matrix}\frac{2q}{3} & \frac{q}{3} & \frac{q}{3} \\ \frac{q}{3} & \frac{2q}{3} & −\frac{q}{3} \\ \frac{q}{3} & −\frac{q}{3} & \frac{2q}{3} \end{matrix} \underrightarrow{C_{1} ← C_{1} − (C_{2} + C_{3})} \begin{matrix}0 & \frac{q}{3} & \frac{q}{3} \\ 0 & \frac{2q}{3} & −\frac{q}{3} \\ 0 & −\frac{q}{3} & \frac{2q}{3} \end{matrix} \underrightarrow{R_{1} ← R_{1} − (R_{2} + R_{3})} \begin{matrix}0 & 0 & 0 \\ 0 & \frac{2q}{3} & −\frac{q}{3} \\ 0 & −\frac{q}{3} & \frac{2q}{3} \end{matrix}

Then C_{2} ⟵ C_{2} − C_{3} gives

\begin{matrix}0 & 0 & 0 \\ 0 & q & −\frac{q}{3} \\ 0 & −q & \frac{2q}{3} \end{matrix} \underrightarrow{R_{3} ← R_{3} + R_{2}} \begin{matrix}0 & 0 & 0 \\ 0 & q & −\frac{q}{3} \\ 0 & 0 & \frac{q}{3} \end{matrix} \underrightarrow{R_{2} ← R_{2} + R_{3}} \begin{matrix}0 & 0 & 0 \\ 0 & q & 0 \\ 0 & 0 & \frac{q}{3} \end{matrix}

Finally, we obtain that N(D′) has Smith normal form

( 1 ; \frac{q}{3} ; \frac{q}{3} ; q ; ... ; q ; 0 ; 0 ; 0 )

so that the invariant factors of the Cartan matrix C_{0}(D′) are d′_{1} = … = d′_{q+1} = 1, d′_{q+2} = d′_{q+3} = 3 and d′_{q+4} = q, and Cart(C′, B_{0}) ≅ Cart(D′, b_{0}) ≅ C_{q} × C_{3} × C_{3}.

3.3.3 Defect and order in the Cartan group

We would like to find some relation between the p-defect d(χ) of a character χ and its order o(χ) in the Cartan group Cart(C), when C is the set of p^{k}- regular elements of the finite group G. We have mentioned that, for any χ ∈ Irr(G), the order o(χ) is a power of p which divides p^{d(χ)}.

We have seen in the previous section that, when G is Sz(q), Re(q) or SU(3, q^{2}) (with q a power of the prime p) and C is the set of p^{2}-regular of G, then all the characters in a given C-block of the principal p-block have the same order in the Cartan group. This is also true when G is Abelian and C is the set of p^{k}-regular elementsof G.

We see that in these particular cases (where all characters in a given Cblock have the same p-defect) the quotient p^{d(χ)}/o(χ) is independant on the choice of χ in a given C-block. We show that this is also true under strong hypotheses:

Theorem 3.3. Let G be a finite group, p be a prime number (dividing |G|), and let C be the set of p^{k}-regular elements of G, for some k ≥ 1. Suppose that B is a C-block of G such that the contribution matrix Γ(C, B) has rank 1 over Q. Then

∀χ, ξ ∈ B, \frac{o(ξ)}{o(χ)} = \frac{p^{d(ξ)}}{p^{d(χ)}} .

Proof. Take χ, ξ ∈ B. Since the rank of Γ(C, B) is 1, we see that there exists a ∈ Q\\{0\} such that ξ^{C} = aχ^{C} (a ≠ 0 since 1 ∈ C). We can write a = \frac{b}{c}, with b, c ∈ Z and gcd(b, c) = 1. Furthermore, we can suppose that gcd(b, p) = 1 (if this is not true, it suffices to exchange the roles of ξ and χ).

Let d be the order of χ in Cart(C, B). Then

∀φ ∈ B, d < χ^{C}, φ^{C} >∈ Z,

and

∃ψ ∈ B such that d < χ^{C}, ψ^{C} >≢ 0 (mod p).

Now, for all φ ∈ B,

d < χ^{C}, φ^{C} > = \frac{da}{a} < χ^{C}, φ^{C} >

= \frac{d}{a} < ξ^{C}, φ^{C} >

= \frac{dc}{b} < ξ^{C}, φ^{C} >∈ Z

and \frac{dc}{b} < ξ^{C}, ψ^{C} >≢ 0 (mod p).

Then, for all φ ∈ B, dc < ξ^{C}, φ^{C} >∈ Z, and, since gcd(b, p) = 1, dc < ξ^{C}, ψ^{C} >≢ 0 (mod p). This implies that the order of ξ in Cart(C, B) is |dc|_{p} = d|c|_{p}, so that |c|_{p} = \frac{d|c|_{p}}{d} = \frac{o(ξ)}{o(χ)}.

On the other hand, 1 ∈ C, so that ξ(1) = aχ(1) = \frac{b}{c}χ(1), and |ξ(1)|_{p} = \frac{|χ(1)|_{p}}{|c|_{p}} (since gcd(b, p) = 1). Hence |c|_{p} = \frac{|χ(1)|_{p}}{|ξ(1)|_{p}}. Therefore

|c|_{p} = \frac{o(ξ)}{o(χ)} = \frac{|χ(1)|_{p}}{|ξ(1)|_{p}} = \frac{p^{d(ξ)}}{p^{d(χ)}} ,

which is the result we wanted.

This result seems to be true in many more cases than the very restrictive situation of the above theorem (cf for example the case of Symmetric groups which we study in the next section). However, it is not true in general, as can be seen for example in the case where G = M_{24} or J_{2} and p^{k} = 4 (this can be shown using GAP on a computer).

3.4 Symmetric Groups

In this section, we study the case of ℓ-blocks of the Symmetric groups. In [18], the authors give the exponent of the Cartan group of an ℓ-block of a given weight, and they conjecture its structure. They also give the determinant of the Cartan matrix of such a block.

We will compute the order of any character in the Cartan group. In all the sequel of this section, we take two integers 2 ≤ ℓ ≤ n, and we let C_{ℓ} be the set of ℓ-regular elements of the symmetric group S_{n}.

3.4.1 Some facts about ℓ-blocks of the symmetric group

Take B an ℓ-block of S_{n} of weight w ≠ 0. The characters of B are labelled by the partitions of n which have a given ℓ-core, γ say. We write B = {χ_{λ}, γ_{λ} = γ}. One of the key ingredients in [18] is that there is a generalized perfect isometry between B and the set of irreducible characters of the wreath product Z_{ℓ}≀S_{w}. One way to see Z_{ℓ}≀S_{w} is to represent it as the set of monomial w by w matrices whose non-zero entries are ℓ-th roots of unity.

We will see that the elements of Irr(Z_{ℓ} ≀ S_{w}) are labelled by ℓ-quotients: for χ_{λ} ∈ B, the quotient β_{λ} is a sequence (λ^{(0)}, λ^{(1)}, …, λ^{(ℓ−1)}) such that, for each 0 ≤ i ≤ ℓ − 1, λ^{(i)} is a partition of some k_{i}, 0 ≤ k_{i} ≤ w, and ∑_{i=0}^{ℓ−1} k_{i} = w (the quotient β_{λ} “stores” the information about λ and how to remove w ℓ-hooks from λ to get γ_{λ}). Each partition is uniquely determined by its ℓ-core and ℓ-quotient.

The subgroup (isomorphic to) S_{w} of Z_{ℓ} ≀ S_{w} acts on the base group Z_{ℓ}^{w} ◁ Z_{ℓ} ≀ S_{w} by conjugation: for (a_{1}, …, a_{w}) ∈ Z_{ℓ}^{w} and σ ∈ S_{w}, we have

σ^{−1}(a_{1}, …, a_{w})σ = (a_{σ(1)}, …, a_{σ(w)}).

We write Irr(Z_{ℓ}) = {α_{0}, …, α_{ℓ−1}}. Then S_{w} also acts by conjugation on Irr(Z_{ℓ}^{w}) via

(α_{i_{1}} ⊗ ⋯ ⊗ α_{i_{w}})^{σ^{−1}} = α_{i_{σ(1)}} ⊗ ⋯ ⊗ α_{i_{σ(w)}},

for all σ ∈ S_{w} and α_{i_{1}} ⊗ ⋯ ⊗ α_{i_{w}} ∈ Irr(Z_{ℓ}^{w}).

The number of conjugacy classes (and of irreducible complex characters) of Z_{ℓ} ≀S_{w} is the number of ℓ-quotients of partitions of n (cf James and Kerber [17], Lemma 4.2.9). In [17], the authors give a description of the conjugacy classes of Z_{ℓ} ≀ S_{w} (cf [17], Theorem 4.2.8). If (a_{1}, …, a_{w})σ ∈ Z_{ℓ} ≀ S_{w}, then (a_{1}, …, a_{w}) can be ”endowed” with a cycle structure, given by that of σ (as it is easy to see in the monomial representation of Z_{ℓ} ≀ S_{w}). Then, for each cycle in (a_{1}, …, a_{w}), the product of the ℓ-th roots of unity corresponding to the a_{i}’s in the cycle is another ℓ-th root of unity. Such a product is called a cycle product in (a_{1}, …, a_{w}). Then the classification of conjugacy classes of Z_{ℓ} ≀ S_{w} is as follows: two elements (a_{1}, …, a_{w})σ and (b_{1}, …, b_{w})τ of Z_{ℓ} ≀ S_{w} are conjugate if and only if σ and τ have the same cycle structure, and, for each cycle length k and each ℓ-th root of unity ζ, the numbers of cycle products equal to ζ and associated to cycles of length k in (a_{1}, …, a_{w}) and (b_{1}, …, b_{w}) respectively are the same.

Now take any χ ∈ Irr(Z_{ℓ} ≀ S_{w}). Then, by Clifford’s Theorem (Isaacs [16], Theorem (6.2)), we have χ|_{Z_{ℓ}^{w}} = e ∑_{i=1}^{t} θ^{σ_{i}} for some integer e and some θ ∈ Irr(Z_{ℓ}^{w}), where {θ^{σ_{i}}, 1 ≤ i ≤ t} is the conjugacy class of θ.

Because of the way S_{w} acts by conjugation on Irr(Z_{ℓ}^{w}), we see that exactly one conjugate of θ has the form α_{0}⊗⋯⊗α_{0}⊗α_{1}⊗⋯⊗α_{1}⊗⋯⊗α_{ℓ−1}⊗⋯⊗ α_{ℓ−1}, where the multiplicity k_{i} of α_{i} is the same as in θ for each 0 ≤ i ≤ ℓ−1. We may assume that θ = α_{0} ⊗ ⋯ ⊗ α_{0} ⊗ ⋯ ⊗ α_{ℓ−1} ⊗ ⋯ ⊗ α_{ℓ−1}.

Again from the action of S_{w} on Irr(Z_{ℓ}^{w}), we see that the inertial subgroup I_{Z_{ℓ}≀S_{w}}(θ) of θ is (Z_{ℓ} ≀ S_{k_{0}}) × ⋯ × (Z_{ℓ} ≀ S_{k_{l−1}}).

Suppose first that I_{Z_{ℓ}≀S_{w}}(θ) = Z_{ℓ} ≀ S_{w}. This means that all the k_{i}’s are 0, expect one. Thus we have θ = α_{i} ⊗ ⋯ ⊗ α_{i} for some 0 ≤ i ≤ ℓ − 1. Using the description we made of the conjugacy classes of Z_{ℓ}≀S_{w}, it is easy to show that (α_{i} ⊗ ⋯ ⊗ α_{i})1_{S_{w}} ∈ Irr(Z_{ℓ} ≀ S_{w}) (where, for ψ ∈ Irr(S_{w}) and (z, σ) ∈ Z_{ℓ} ≀ S_{w}, we let ((α_{i} ⊗ ⋯ ⊗ α_{i})ψ)((z, σ)) = (α_{i} ⊗ ⋯ ⊗ α_{i})(z)ψ(σ)). One first shows, using the description above, that (α_{i} ⊗ ⋯ ⊗ α_{i})1_{S_{w}} is a class function of Z_{ℓ} ≀S_{w}; then it is clear that < (α_{i} ⊗⋯⊗α_{i})1_{S_{w}}, (α_{i} ⊗⋯⊗α_{i})1_{S_{w}} >_{Z_{ℓ}≀S_{w}}= 1, so that (α_{i} ⊗⋯⊗α_{i})1_{S_{w}} ∈ Irr(Z_{ℓ} ≀S_{w}). Furthermore, (α_{i} ⊗⋯⊗α_{i})1_{S_{w}}|_{Z_{ℓ}^{w}} = (α_{i} ⊗ ⋯ ⊗ α_{i}). Thus, by a result by Gallagher ([16], Corollary (6.17)), the characters (α_{i}⊗⋯⊗α_{i})ψ for ψ ∈ Irr(S_{w}) are irreducible, distinct for distinct ψ, and are all the irreducible characters of Z_{ℓ}≀S_{w} which lie over (α_{i}⊗⋯⊗α_{i}). In particular, χ has the form (α_{i}⊗⋯⊗α_{i})χ_{λ} for some λ ⊢ w, and i and χ_{λ} are uniquely determined by χ. We label χ by the quotient (λ^{(0)}, λ^{(1)}, …, λ^{(ℓ−1)}) where λ^{(i)} = λ and, for each 0 ≤ i ≠ j ≤ ℓ − 1, λ^{(j)} = ∅.

Now suppose I_{Z_{ℓ}≀S_{w}}(θ) < Z_{ℓ} ≀ S_{w}. Then, by Clifford’s Theory ([16], Theorem 6.11), χ is induced by a unique irreducible character of I_{Z_{ℓ}≀S_{w}}(θ) which lies over θ. By the same argument as above, we see that such a character must have the form (α_{0} ⊗ ⋯ ⊗ α_{0})χ_{λ^{(0)}} ⊗ ⋯ ⊗ (α_{ℓ−1} ⊗ ⋯ ⊗ α_{ℓ−1})χ_{λ^{(ℓ−1)}} where, for each 0 ≤ i ≤ ℓ − 1, λ^{(i)} ⊢ k_{i}. And, here again, the sequence (χ_{λ^{(0)}}, …, χ_{λ^{(ℓ−1)}}) is uniquely determined by χ. We therefore label χ by the quotient (λ^{(0)}, …, λ^{(ℓ−1)}).

We see that, in this way, as χ runs through Irr(Z_{ℓ} ≀S_{w}), all the ℓ-quotients appear (once). Thus, to each quotient β_{λ} can be associated a uniquely determined irreducible character of Z_{ℓ} ≀ S_{w}, which we denote by χ_{β_{λ}}.

From the description of χ_{β_{λ}} we gave, we compute easily its degree (using the fact that the α_{i}’s have degree 1). We have, writing θ_{λ} for θ and η_{λ} for ⨂_{i=0}^{ℓ−1}(α_{i} ⊗ ⋯ ⊗ α_{i})χ_{λ^{(i)}},

χ_{β_{λ}}(1) = (Ind_{I_{Z_{ℓ}≀Sw} (θ_{λ})}^{Z_{ℓ}≀S_{w}}(η_{λ}))(1)

= \frac{|Z_{ℓ} ≀ S_{w}|}{|I_{Z_{ℓ}≀S_{w}}(θ_{λ})|}η_{λ}(1)

= \frac{ℓ^{w}w!}{∏_{i=0}^{ℓ−1} ℓ^{k_{i}}k_{i}!}χ_{λ^{(0)}}(1) … χ_{λ^{(ℓ−1)}}(1)

= \frac{w!}{∏_{i} k_{i}!} ∏_{i=0}^{ℓ−1} \frac{k_{i}!}{∏ hook lengths in λ^{(i)}}

(this last equality being true because of the Hook Length Formula (cf [17])). Hence

χ_{β_{λ}}(1) = \frac{w!}{∏ hook lengths in β_{λ}}

(where we call hook in β_{λ} a hook in any of the partitions appearing in β_{λ}), and

\frac{|Z_{ℓ} ≀ S_{w}|}{χ_{β_{λ}}(1)} = ℓ^{w} ∏ hook lengths in β_{λ}. (††)

The key point for the reductions in [18] and in the end of this section is the following: the map χ_{λ} ⟼ χ_{β_{λ}} is a generalized perfect isometry between B and Irr(Z_{ℓ} ≀S_{w}) with respect to ℓ-regular elements of S_{n} and regular elements of Z_{ℓ}≀S_{w} (where an element of Z_{ℓ}≀S_{w} is regular if 1 is not an eigenvalue of the corresponding monomial matrix). This is proved in [18]. For χ_{λ}, χ_{μ} ∈ B, there exist signs σ_{λ} and σ_{μ} such that < χ_{λ}, χ_{μ} >_{C_{ℓ}}=< σ_{λ}χ_{β_{λ}}, σ_{μ}χ_{β_{μ}} >_{reg} (where < , >_{C_{ℓ}} and < , >_{reg} denote the scalar products across ℓ-regular elements of S_{n} and across regular elements of Z_{ℓ}≀S_{w} respectively). In particular, χ_{λ} and χ_{β_{λ}} have the same order in the corresponding Cartan groups (“note” that the set of regular elements in Z_{ℓ} ≀S_{w} is a closed set of conjugacy classes).

Note that an ℓ-block B of S_{n} of weight 0 consists of a single character. We have B = {χ_{λ}} where λ is its own ℓ-core. In this case, we have Z_{ℓ} ≀ S_{w} = {1}, and we define reg, the set of regular elements of Z_{ℓ} ≀ S_{0} to be {1}. We have Irr(Z_{ℓ} ≀ S_{w}) = {1_{Z_{ℓ}≀S_{w}}}. Now χ_{λ} vanishes outside C_{ℓ} (by Proposition 3.1), so that χ_{λ}^{C_{ℓ}} = χ_{λ}, and we again get a generalized perfect isometry between B and Irr(Z_{ℓ} ≀ S_{0}) (with respect to C_{ℓ} and reg). The corresponding Cartan groups Cart(ℓ, B) and Cart(ℓ, 0) are both (isomorphic to) the trivial group C_{1}.

3.4.2 Invariant factors of the Cartan matrix

The following results can be found in [18]. Let B be an ℓ-block of S_{n} of weight w. Then there is a generalized perfect isometry between B Irr(Z_{ℓ} ≀S_{w})

(with respect to ℓ-regular classes in S_{n} and regular classes in Z_{ℓ} ≀ S_{w}). This implies that the corresponding Cartan groups, Cart(C_{ℓ}, B) = Cart(ℓ, B) and Cart(reg, Z_{ℓ} ≀ S_{w}) = Cart(ℓ, Z_{ℓ} ≀ S_{w}) = Cart(ℓ, w), are isomorphic, and thus that the Cartan matrices C(ℓ, B) and C(ℓ, w) have the same invariant factors. This allows the authors to work in Z_{ℓ} ≀ S_{w}, and then transfer the results back to B.

In the course of their work, they show that every irreducible character of Z_{ℓ} ≀ S_{w} is directly linked across regular elements to the trivial character (cf [18], Theorem 5.12). More precisely, we have

∀χ ∈ Irr(Z_{ℓ} ≀ S_{w}), Z ∋ \frac{ℓ^{w}w! < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}}{χ(1)} ≡ (−1)^{w} (mod ℓ).

Writing π the set of primes dividing ℓ, every positive integer m factors uniquely as m = m_{π}m_{π′}, where every prime factor of m_{π} belongs to π and no prime factor of m_{π′} is contained in π. We call m_{π} the π-part of m. It can be shown (cf Donkin [10]) that Cart(ℓ, w) is a π-group. Together with the above result, this gives the following

Theorem 3.4. ([18] Theorem 6.2 and Corollary 6.3) The exponent of Cart(ℓ, w) (i.e. the order of 1_{Z_{ℓ}≀S_{w}} in Cart(ℓ, w)) is ℓ^{w}w!_{π}. This in turns implies that the exponent of Cart(C_{ℓ}, S_{n}) is ℓ^{⌊ \frac{n}{ℓ} ⌋}⌊\frac{n}{ℓ}⌋!_{π}.

We next turn to the invariant factors of the C_{ℓ}-Cartan matrix C_{n} of S_{n}. A partition λ of n is called an ℓ-class regular partition of n if no part of λ is divisible by ℓ. We then write λ ⊢_{ℓ} n. If m is a positive integer, we define ℓ_{m} = \frac{ℓ}{gcd(ℓ,m)}, and let π_{m} be the set of primes dividing ℓ_{m}. If a is also a positive integer, we let

r_{ℓ}(m, a) = ℓ_{m}^{⌊ \frac{a}{ℓ} ⌋}⌊\frac{a}{ℓ}⌋!_{π_{m}}.

Finally, if λ ⊢_{ℓ} n is written exponentially λ = (1^{a_{1}(λ)}, 2^{a_{2}(λ)}, …), we let

r_{ℓ}(λ) = ∏_{m≥1} r_{ℓ}(m, a_{m}(λ)).

We then have the following

Conjecture 3.5. ([18] Conjecture 6.4) The C_{ℓ}-Cartan group Cart(ℓ, S_{n}) is a direct product of cyclic groups of order r_{ℓ}(λ), where λ runs through the set of ℓ-class regular partitions of n. In particular, the determinant of an ℓ-Cartan matrix C_{n} of S_{n} is

det(C_{n}) = ∏_{λ⊢_{ℓ}n} r_{ℓ}(λ).

This conjecture is supported by a number of examples computed by the authors. Furthermore, they show that the consequence they mention is true:

Theorem 3.6. ([18] Theorem 6.13) We have

det(C_{n}) = ∏_{λ⊢_{ℓ}n} r_{ℓ}(λ).

3.4.3 Orders in the Cartan group

We now compute, for an ℓ-block B of S_{n}, the orders of the characters of B in the Cartan group Cart(ℓ, B). We will need the following three lemmas:

Lemma 3.7. If m ≥ 1 and λ, partition of n, has m-weight w, then there are w hooks in λ whose length is divisible by m.

Lemma 3.8. For any 0 ≠ m ∈ N and partition λ, there is a bijection between the set of m-hooks in β_{λ} (the ℓ-quotient of λ) and the set of mℓ-hooks in λ (where a m-hook (resp. mℓ-hook) is a hook whose length is exactly m (resp. mℓ)).

These two results can be seen by using the abacus, as presented in [17].

Lemma 3.9. If n ∈ N, 2 ≤ ℓ ≤ n, and if m|ℓ for some m ≥ 2, then any m-block of S_{n} is a union of ℓ-blocks.

Proof. Two proofs can be given:

1. Write ℓ = mr. Suppose χ_{λ}, χ_{μ} ∈ Irr(S_{n}) belong to the same ℓ-block. Then λ and μ have the same ℓ-core, and this can be obtained by removing from λ and μ a certain number of ℓ-hooks (this number being the weight of the ℓ-block). However, the removal of an ℓ-hook, i.e. an mr-hook, can be obtained by removing r m-hooks (cf for example the proof of Lemma 3.2 in [18]). Hence the common ℓ-core γ of λ and μ can be obtained by removing m-hooks, and, removing m-hooks from γ, one must eventually obtain an mcore, which will thus be the same for λ and μ. Hence χ_{λ} and χ_{μ} belong to the same m-block, giving the result.

2. It is “easy” to see from the definitions that any ℓ-cycle section is a union of m-cycle sections (for, if the m-cycle parts of α, β ∈ S_{n} are conjugate, then certainly their ℓ-cycle parts are conjugate too. Hence, if α and β belong to the same m-cycle section, then they belong to the same ℓ-cycle section).

Now suppose χ, ψ ∈ Irr(S_{n}) belong to distinct m-blocks. Then, the mblocks of S_{n} having the Second Main Theorem Property, χ and ψ are orthogonal across each m-cycle section of S_{n}. By the above remark, we see that χ and ψ are thus also orthogonal across each ℓ-cycle section, in particular C_{ℓ}, and thus belong to distinct ℓ-blocks. A priori, we just get that χ and ψ are not directly linked across ℓ-regular elements. But then, if χ_{0} = χ, χ_{1}, …, χ_{r} is a sequence of characters such that χ_{i} is directly C_{ℓ}- linked to χ_{i+1} for 0 ≤ i ≤ r − 1, then, by the above, all the χ_{i}’s must lie in the same m-block. In particular, the ℓ-block containing χ is contained in the m-block containing χ. Hence the result.

Some notations: We take 2 ≤ ℓ ≤ n ∈ N. For λ a partition of n, we will write HL_{λ} (or just HL when the context is clear) for a hook length in λ. For any integer m dividing ℓ, we will write (m) − HL for a hook length divisible by m, and {m} − HL for a hook length whose π-part is precisely m (where π is the set of primes dividing ℓ). With a slight abuse of notation, we define similarly HL_{β_{λ}} to be a hook length in any of the partitions appearing in the quotient β_{λ}. Finally, a hook whose length is divisible by m will be called an (m)-hook.

Definition 3.10. For any (finite) group G and χ ∈ Irr(G), and for ℓ ≥ 2 an integer dividing |G|, writing π the set of primes dividing ℓ, we define the π-defect d_{π}(χ) of χ, via

d_{π}(χ) = (d_{p_{1}}(χ), …, d_{p_{s}}(χ))

where π = {p_{1}, …, p_{s}} and, for 1 ≤ i ≤ s, d_{p_{i}}(χ) is the p_{i}-defect of χ. We write (abusively)

π^{d_{π}(χ)} := ∏_{i=1}^{s} p_{i}^{d_{p_{i}}(χ)} = \frac{|G|_{π}}{χ(1)_{π}}

(where |G|_{π} and χ(1)_{π} are the π-parts of |G| and χ(1) respectively).

We first establish the following:

Proposition 3.11. If 2 ≤ ℓ ≤ n, if B is an ℓ-block of S_{n} of weight w ≠ 0, if χ_{λ} ∈ B and if χ_{λ} corresponds to χ_{β_{λ}} under one of the perfect isometries we described between B and Irr(Z_{ℓ} ≀ S_{w}), then

π^{d_{π}(χ_{β_{λ}})} = (∏(ℓ) − HL_{λ})_{π}.

Proof. First note that (††) implies that

π^{d_{π}(χ_{β_{λ}})} = (ℓ^{w} ∏ HL_{β_{λ}})_{π}.

Now there are precisely w hooks in β_{λ}, so that

π^{d_{π}(χ_{β_{λ}})} = (∏ ℓ.HL_{β_{λ}})_{π}.

On the other hand, by applying Lemma 3.7 for each hook length appearing in β_{λ}, we see that there is a bijection between the set of hooks in β_{λ} and the set of hooks of length divisible by ℓ in λ (which is, in fact, the statement of Lemma 3.6), and that, if two hooks correspond to each other under this bijection, then HL_{λ} = ℓ.HL_{β_{λ}}. Hence

∏ ℓ.HL_{β_{λ}} = ∏(ℓ) − HL_{λ}.

Taking π-parts, we finally obtain

π^{d_{π}(χ_{β_{λ}})} = (∏(ℓ) − HL_{λ})_{π}.

We next prove a result in the special case where ℓ is a prime power. It is this proposition which will in the end give us the result we announced in the previous section (i.e. that the quotient of the defect of a character by its order in the Cartan group is an invariant of the block).

Proposition 3.12. Let p be a prime and k ≥ 1 be such that 2 ≤ p^{k} ≤ n. If B is a p^{k}-block of S_{n} and if χ_{λ}, χ_{μ} ∈ B, then

\frac{(∏(p^{k}) − HL_{λ})_{p}}{(∏(p^{k}) − HL_{μ})_{p}} = \frac{p^{d_{p}(χ_{λ})}}{p^{d_{p}(χ_{μ})}} .

Proof. By Lemma 3.8, for all 1 ≤ i ≤ k, χ_{λ} and χ_{μ} belong to the same p^{i}-block of S_{n}. Thus, by Lemma 3.6, λ and μ have the same number of (p^{i})-hooks for each 1 ≤ i ≤ k.

Now, if 1 ≤ i ≤ k − 1, λ and μ have the same number of (p^{i})-hooks and the same number of (p^{i+1})-hooks. Since the set of (p^{i})-hooks is the disjoint union of the set of (p^{i+1})-hooks and the set of {p^{i}}-hooks, we see that λ and μ also have the same number of {p^{i}}-hooks (note that, by definition, for such a hook, the p-part of the hook length is exactly p^{i}).

By the Hook Length Formula, we have that

p^{d_{p}(χ_{λ})} = (∏ HL_{λ})_{p} = (∏(p) − HL_{λ})_{p}.

Thus

\begin{aligned}p^{d_{p}(χ_{λ})} & = (∏_{i=1}^{k−1} ∏(\{p^{i}\} − HL_{λ})_{p}) (∏(p^{k}) − HL_{λ})_{p} \\ & = (∏_{i=1}^{k−1} p^{i|\{\{p^{i}\}-hooks in λ\}|}) (∏(p^{k}) − HL_{λ})_{p} \end{aligned}

And, since, for all 1 ≤ i ≤ k − 1, |{{p^{i}}-hooks in λ}| = |{{p^{i}}-hooks in μ}|, it is easy to see that

\frac{p^{d_{p}(χ_{λ})}}{p^{d_{p}(χ_{μ})}} = \frac{(∏(p^{k}) − HL_{λ})_{p}}{(∏(p^{k}) − HL_{μ})_{p}} .

We now turn to the actual computation of the order of an irreducible character of the wreath product in the Cartan group. It turns out that it suffices to know the product across regular classes with the trivial character, and this has been computed in [18].

Proposition 3.13. Take 2 ≤ ℓ ∈ N and 0 ≠ w ∈ N. If χ ∈ Irr(Z_{ℓ}≀S_{w}), then

the order o(χ) of χ in the Cartan group Cart(ℓ, Z_{ℓ} ≀ S_{w}) is \frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} = π^{d_{π}(χ)}.

Proof. Take χ ∈ Irr(Z_{ℓ} ≀ S_{w}). Recall that

Z ∋ \frac{ℓ^{w}w!}{χ(1)} < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}≡ (−1)^{w} (mod ℓ) (•)

Now o(χ) is a π-number, so that < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg} is a rational whose (reduced) denominator is a π-number. This implies that

\frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}∈ Z.

Furthermore, from (•), we also deduce that, for each p ∈ π,

\frac{ℓ^{w}w!}{χ(1)} < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}≢ 0 (mod p).

Thus, for any p ∈ π,

\frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}≢ 0 (mod p).

Hence \frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} is the smallest positive integer d such that d < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}∈

Z. This implies that

\frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} |o(χ)

(indeed, by definition, o(χ) < χ, 1_{Z_{ℓ}≀S_{w}} >_{reg}∈ Z, and o(χ) is a π-number).

Now, conversely, if ψ ∈ Irr(Z_{ℓ} ≀ S_{w}), then < χ, ψ >_{reg}∈ Q, so also (since χ(1) divides |Z_{ℓ} ≀ S_{w}| = ℓ^{w}w!)

\frac{ℓ^{w}w!}{χ(1)} < χ, ψ >_{reg}∈ Q.

However,

\frac{ℓ^{w}w!}{χ(1)} < χ, ψ >_{reg}= \frac{ℓ^{w}w!}{ℓ^{w}w!} ∑_{g∈reg/∼} \frac{K_{g}χ(g)}{χ(1)}ψ(g^{−1})

(where the sum is taken over representatives for the regular classes, and, for g such a representative, K_{g} is the size of the conjugacy class of g). And, for each g in the sum, \frac{K_{g}χ(g)}{χ(1)} and ψ(g^{−1}) are both algebraic integers. Hence

\frac{ℓ^{w}w!}{χ(1)} < χ, ψ >_{reg} is also an algebraic integer, and thus an integer. Thus

∀ψ ∈ Irr(Z_{ℓ} ≀ S_{w}), \frac{ℓ^{w}w!}{χ(1)} < χ, ψ >_{reg}∈ Z

and this implies that o(χ) divides \frac{ℓ^{w}w!}{χ(1)}, and, o(χ) being a π-number,

o(χ)|\frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} .

Hence we finally get

o(χ) = \frac{ℓ^{w}(w!)_{π}}{χ(1)_{π}} = π^{d_{π}(χ)}.

Putting together the results of these three propositions, we obtain the following

Theorem 3.14. Let 2 ≤ ℓ ≤ n, and let B be an ℓ-block of S_{n} of weight w. Then

• (i) If w = 0, then B = {χ_{λ}} for some partition λ of n, and χ_{λ} has order 1 in the Cartan group Cart(ℓ, S_{n}).

• (ii) If w ≠ 0, and if χ_{λ} ∈ B, then the order of χ_{λ} in Cart(ℓ, S_{n}) is (∏(ℓ) − HL_{λ})_{π}, where π is the set of primes dividing ℓ (i.e. o(χ_{λ}) is the π-part of the product of the hook lengths divisible by ℓ in λ).

• (iii) If ℓ = p^{k} for some prime p and k ≥ 1, and if χ, ψ ∈ B, then

\frac{o(χ)}{o(ψ)} = \frac{p^{d_{p}(χ)}}{p^{d_{p}(ψ)}} .

106 PART 3. CARTAN GROUP, GENERALIZED CHARACTERS

Finite General Linear Group

In all this part, we let n be a positive integer and q be a power of a prime p. We will work in the finite general linear group G, which can be seen as the group GL(V) of automorphisms of an n-dimensional vector space V over the finite field F_{q}, or as its natural matrix representation GL(n, q), the group of invertible n by n matrices with entries in F_{q}. The irreducible (complex) characters of G have been described by Green in [15], using deep combinatorial arguments. Then, using in particular the Deligne-Lusztig theory, Fong and Srinivasan have classified the blocks of G (cf [12]). They show in particular that unipotent characters behave nicely with respect to blocks. The unipotent characters of GL(n, q) are parametrized by the partitions of n. It turns out that, if r is an odd prime not dividing q, then two unipotent characters belong to the same r-block of G if and only if the partitions labelling them have the same e-core, where e is the multiplicative order of q modulo r. This result is shown using analogues of the Murnaghan-Nakayama rule for irreducible characters of G. Our aim is to use these analogues to obtain properties of some generalized blocks we define in G. So far, we did this only for unipotent irreducible characters of G. However, we have good hope that this could be generalized to arbitrary irreducible characters. We also believe that the same thing should work in the unitary groups. We construct unipotent blocks for G which satisfy one sense of an analogue of the Nakayama Conjecture. Unions of these unipotent blocks then satisfy this analogue, and they have the Second Main Theorem property.

For a more general survey of the representation theory of finite reductive groups, we refer to Cabanes and Enguehard [5].

4.1 Conjugacy classes

4.1.1 Rational canonical form

We first introduce the theory of elementary divisors and the rational canonical form in G, which give a parametrization of the conjugacy classes. We present this in the matrix representation GL(n, q) of G, where things are seen more easily. For the results we give in this section, we refer to Green [15].

The conjugacy classes of G are parametrized by the sequences (… f_{i}^{ν_{i}} …), where F = {f_{i}, i ∈ I} is the set of irreducible, monic polynomials distinct from X and of degree at most n in F_{q}[X] (so that I is finite), and the ν_{i}’s are partitions of non-negative integers k_{i} such that ∑_{i} k_{i} = n. We will write ν ⊢ k to say that ν is a partition of k, and we will write |ν| = k.

In all the sequel, it will be assumed that, when we write (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}) for a conjugacy class, then polynomials which don’t contribute have been omitted (i.e. |ν_{1}| ≠ 0, …, |ν_{r}| ≠ 0).

Take a conjugacy class c = (… f_{i}^{ν_{i}} …) = (… f_{i}^{ν(f_{i})} …) of G, and g ∈ c. Then, the characteristic polynomial of g over F_{q} is

Char(g) = ∏_{i∈I} f_{i}^{k_{i}}.

Writing, for each i, ν(f_{i}) = ν_{i} = (λ_{i,1} ≥ λ_{i,2} ≥ … ≥ λ_{i,s}) (where s can be chosen to be big enough for all f′_{i}s appearing in Char(g) (and even for all g ∈ G), by taking for example s = n), the minimal polynomial of g over F_{q} is

Min(g) = ∏_{i∈I} f_{i}^{λ_{i,1}}.

For k = 1, …, s, the polynomial ED_{k}(g) = ∏_{i} f_{i}^{∑_{j=1}^{k} λ_{i,j}} is the k-th elementary divisor of g over F_{q}.

For any irreducible monic polynomial f over F_{q}, we let U(f) be the companion matrix of f, i.e. if f(X) = X^{d} − a_{d−1}X^{d−1} − ⋯ − a_{0}, then

U(f) = U_{1}(f) = ( \begin{matrix}0 & 1 \\ ⋮ & ... & ... & (0) \\ ⋮ & & ... & ... \\ 0 & ⋯ & ⋯ & 0 & 1 \\ a_{0} & a_{1} & ⋯ & ⋯ & a_{d−1} \end{matrix} ) .

For any positive integer λ, we write

U_{λ}(f) = ( \begin{matrix}U(f) & I_{d} \\ & U(f) & I_{d} \\ & & ... & ... \\ & & & ... & I_{d} \\ & & & & U(f) \end{matrix} ) ,

where I_{d} is the d by d identity matrix, and there are λ blocks U(f) on the diagonal. Note that U_{λ}(f) is equivalent over F_{q} to the companion matrix of f^{λ}. Finally, for ν = (λ_{1} ≥ … ≥ λ_{s}) any partition (of an integer k say), we write

U_{ν}(f) = ( \begin{matrix}U_{λ_{1}}(f) \\ & . & . & . \\ & & & & U_{λ_{s}}(f) \end{matrix} ) .

If g ∈ c = (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}), then the matrix of g is equivalent over F_{q} to

U_{(f_{1}^{ν_{1}}, ..., f_{r}^{νr})} = ( \begin{matrix}U_{ν_{1}}(f_{1}) \\ & . & . & . \\ & & & & U_{ν_{r}}(f_{r}) \end{matrix} ) .

This is the rational canonical form of g over F_{q}.

Note that the theory of elementary divisors is independant on the representation of G we take, so that we don’t change anything by, for example, taking G = GL(V) and taking any basis for V.

4.1.2 The Jordan decomposition

An element g ∈ G = GL(n, q) is semisimple if and only if it is diagonalizable over an algebraic extension of F_{q} (i.e. some F_{q^{k}}, k ≥ 1), i.e. Min(g) splits over F_{q^{k}} and has only simple roots. Let c be the conjugacy class of g. If c = (… f^{ν(f)} …), then g is semisimple if and only if for any f appearing in c (i.e. |ν(f)| ≠ 0), ν(f) = (1, …, 1). We write Δ((q_{i})) for the diagonal matrix with diagonal blocks the q_{i}’s. Suppose g = Δ((U_{ν_{i}}(f_{i}))). Then g = g_{S} +g_{N} = Δ((Uν̃_{i}(f_{i}))) + Δ((Ĩ_{ν_{i}}(f_{i}))), where ν̃_{i}(f_{i}) = (1, …, 1) ⊢ |ν_{i}(f_{i})| = k_{i} and

Ĩ_{ν_{i}}(f_{i}) = ( \begin{matrix}(0) & I_{d_{i}} \\ & (0) & I_{d_{i}} \\ & & ... & ... \\ & & & ... & I_{d_{i}} \\ & & & & (0) \end{matrix} ) ,

with |ν_{i}| = k_{i} diagonal d_{i} by d_{i} blocks, where f_{i} has degree d_{i}.

Thus g_{S} = Δ((U_{k_{i}}(f_{i}))) is semisimple and commutes with g_{N}, which is nilpotent. Hence g_{S} is the semisimple part of g (cf e.g. the introduction of Carter [6]). The unipotent part of g is g_{U} = I_{n} + g_{S}^{−1}g_{N}.

Now, if g̃ = hgh^{−1} for some h ∈ G, then g̃ = (hg_{S}h^{−1})(hg_{U}h^{−1}), and we see that the semisimple and unipotent parts of g̃ are hg_{S}h^{−1} and hg_{U}h^{−1} respectively.

4.1.3 Primary decomposition

Now, we consider G as the group of automorphisms GL(V) of an n-dimensional vector space over F_{q}. We first mention a fact about centralizers in GL(V). Take g ∈ GL(V). If, in some matrix representation corresponding to the decomposition V = V_{1} ⊕ V_{2}, we have

g = ( \begin{matrix}g_{1} \\ & g_{2} \end{matrix} )

with, for i ∈ {1, 2}, g_{i} ∈ G_{i} = GL(V_{i}), and gcd(Min(g_{1}), Min(g_{2})) = 1, then

C_{G}(g) = \{( \begin{matrix}a \\ & b \end{matrix} ) , a ∈ C_{G_{1}}(g_{1}), b ∈ C_{G_{2}}(g_{2})\} ,

and thus C_{G}(g) ≅ C_{G_{1}}(g_{1}) × C_{G_{2}}(g_{2}) via h ⟼ (h|_{V_{1}}, h|_{V_{2}}) (which is independant on the matrix representation).

We also give the following lemma

Lemma 4.1. Let g ∈ G = GL(V). Suppose V_{1} and V_{2} are g-stable subspaces of V such that V = V_{1} ⊕ V_{2}. If g|_{V_{1}} ∈ (… f^{ν_{1}(f)} …)_{f∈F} and g|_{V_{2}} ∈ (… f^{ν_{2}(f)} …)_{f∈F}, then g ∈ (… f^{ν_{1}(f)∪ν_{2}(f)} …)_{f∈F}, where, by ∪, we denote the concatenation of partitions (i.e. the components of ν_{1} ∪ ν_{2} are those of ν_{1} together with those of ν_{2}. If ν_{1} ⊢ n_{1} and ν_{2} ⊢ n_{2}, then ν_{1} ∪ ν_{2} ⊢ n_{1} + n_{2}.)

Proof. Taking any bases for V_{1} and V_{2} (which then add to give a basis of V), we can write

g = ( \begin{matrix}g_{1} \\ & g_{2} \end{matrix} )

(where g_{1} (resp. g_{2}) is the matrix of g|_{V_{1}} (resp. g|_{V_{2}})). The idea is that we can obtain the rational canonical form of g by reducing to this form g_{1} and g_{2}. There exist h_{1} ∈ GL(V_{1}) and h_{2} ∈ GL(V_{2}) such that

g = ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} )^{−1} ( \begin{matrix}U_{(...f^{ν_{1}(f)}...)} \\ & U_{(...f^{ν_{2}(f)}...)} \end{matrix} ) ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} ) .

4.2. (X , Y)-SECTIONS 111

Thus, for some permutation matrix P,

g = ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} )^{−1} P^{−1}U_{(...f^{ν_{1}(f)∪ν_{2}(f)}...)}P ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} ) ,

so that g ∈ (… f^{ν_{1}(f)∪ν_{2}(f)} …)_{f∈F}.

We now turn to the primary decomposition of elements of G. Take g ∈ GL(V) and suppose g ∈ (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}). Then there exists a unique decomposition V = V_{1} ⊕ ⋯ ⊕ V_{r}, where the V_{i}’s are g-stable subspaces of V and, for 1 ≤ i ≤ r, g|_{V_{i}} ∈ (f_{i}^{ν_{i}}) ⊂ GL(V_{i}). For each 1 ≤ i ≤ r, V_{i} is given by

V_{i} = {v ∈ V | f_{i}^{k}(g)v = 0 for some k > 0}.

We have C_{G}(g) ≅ C_{GL(V_{1})}(g|_{V_{1}}) × ⋯ × C_{GL(V_{r})}(g|_{V_{r}}). Then there exists a unique writing g = g_{1} … g_{r}, where, for each 1 ≤ i ≤ r, g_{i} ∈ GL(V), V_{i} is g_{i}-stable and g_{i}|_{V_{j}} = 1 for all j ≠ i. Indeed, we must have g_{i}|_{V_{i}} = g|_{V_{i}} for each 1 ≤ i ≤ r, and g_{i} is uniquely determined by this and its properties listed before. Furthermore, the g_{i}’s are pairwise commuting elements. We say that g_{1} … g_{r} is the primary decomposition of g. The g_{i}’s are the primary components of g.

More generally, an element of G is said to be primary if its characteristic polynomial is divisible by at most one irreducible polynomial distinct from X − 1. We have the following (cf Fong-Srinivasan [12])

Proposition 4.2. Suppose h is a semisimple primary element of some general linear group GL(m, q), and that h ∈ (f^{ν}) for some f ≠ X − 1. Writing d the degree of f, we thus have m = kd where ν = (1, …, 1) ⊢ k. Then

C_{GL(kd,q)}(h) ≅ GL(k, q^{d}).

From the primary decomposition of g ∈ G, we deduce the following: if F is the disjoint union of F_{1} and F_{2}, then there is a unique decomposition V = V_{1} ⊕V_{2} where V_{1} and V_{2} are g-stable, g|_{V_{1}} ∈ (… f^{ν(f)} …)_{f∈F_{1}} and g|_{V_{2}} ∈ (… f^{ν(f)} …)_{f∈F_{2}}. Then g has a unique decomposition g = g_{1}g_{2} = g_{2}g_{1} where g_{1}, g_{2} ∈ GL(V), g_{1}|_{V_{2}} = 1 and g_{2}|_{V_{1}} = 1. Indeed, under these hypotheses, necessarily, for i ∈ {1, 2}, V_{i} is g_{i}-stable and g_{i}|_{V_{i}} = g|_{V_{i}}. We then have C_{G}(g) ≅ C_{GL(V_{1})}(g|_{V_{1}}) × C_{GL(V_{2})}(g|_{V_{2}}).

4.2 (X , Y)-sections

We will give four definitions of sections in G. The first two will give us information on blocks defined only on the unipotent characters of G. The other two give us bigger sections, and thus smaller blocks, but they should allow us to work on the whole of Irr(G).

The idea of the following definitions is, in the rational canonical form we gave above, to isolate blocks corresponding to irreducible polynomials whose degree is equal to or divisible by a given d (and which, in the last two definitions, are semisimple). We let, writing δ(f) the degree of any polynomial f,

F_{d} = {f ∈ F | f ≠ X − 1 and δ(f) = d} , F_{d}^{0} = F \ F_{d}
F_{(d)} = {f ∈ F | f ≠ X − 1 and d|δ(f)} and F_{(d)}^{0} = F \ F_{(d)}.

We define four unions of conjugacy classes of G:

X_{d} = {(… f^{ν(f)} …) | f ∈ F_{d} ∪ {X − 1} and ν(X − 1) = (0) or (1, …, 1)},

X_{(d)} = {(… f^{ν(f)} …) | f ∈ F_{(d)} ∪ {X − 1} and ν(X − 1) = (0) or (1, …, 1)},

X_{d}^{(s)} = {(… f^{ν(f)} …) | f ∈ F_{d} ∪ {X − 1} and ν(f) = (0) or (1, …, 1)},

X_{(d)}^{(s)} = {(… f^{ν(f)} …) | f ∈ F_{(d)} ∪ {X − 1} and ν(f) = (0) or (1, …, 1)}

(Then X_{d}^{(s)} (resp. X_{(d)}^{(s)}) is the set of semisimple elements of X_{d} (resp. X_{(d)}).)

We have 1 ∈ X_{d} ∩ X_{(d)} ∩ X_{d}^{(s)} ∩ X_{(d)}^{(s)}.

For x ∈ X_{d} (resp. x ∈ X_{(d)}), we let F_{x} = F_{d} and F_{x}^{0} = F_{d}^{0} (resp. F_{x} = F_{(d)} and F_{x}^{0} = F_{(d)}^{0}). For x ∈ X_{d}^{(s)} (resp. x ∈ X_{(d)}^{(s)}), we let F_{x} = {f ∈ F_{d} | ν_{x}(f) ≠ (0)} (resp. F_{x} = {f ∈ F_{(d)} | ν_{x}(f) ≠ (0)}) and F_{x}^{0} = F \ F_{x}.

We let X = X_{d}, X_{(d)}, X_{d}^{(s)} or X_{(d)}^{(s)}. Then, for each x ∈ X, the set F is the disjoint union of F_{x} and F_{x}^{0}. For each x ∈ X, there exists a unique decomposition V = V_{x} ⊕ V_{x}^{0} such that V_{x} is x-stable, x|_{V_{x}^{0}} = 1 and x|_{V_{x}} ∈ (… f^{ν(f)} …)_{f∈F_{x}}. We then have C_{G}(x) ≅ C_{GL(V_{x})}(x|_{V_{x}}) × GL(V_{x}^{0}). We let

Y(x) = {y ∈ GL(V) | V_{x}^{0} is y -stable, y|_{V_{x}} = 1 and y|_{V_{x}^{0}} ∈ (… f^{ν(f)} …)_{f∈F_{x}^{0}}}.

From the definitions, we see that Y(x) ⊂ C_{G}(x) for any x ∈ X. We also see, using the remarks we made on the primary decomposition, that, for any g ∈ G, there exists unique x ∈ X, y ∈ Y(x) such that g = xy. Indeed, if g ∈ (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}) and if we write, as above, V = V_{1}⊕⋯⊕V_{r} and g = g_{1} … g_{r}, then we necessarily have V_{x} = ⊕_{i∈I}V_{i}, where I ⊂ {1, …, r} is the set of indices i such that f_{i}^{ν_{i}} has the property defining X, V_{x}^{0} = ⊕_{i∉I}V_{i}, x|_{V_{x}} = g|_{V_{x}} and y|_{V_{x}^{0}} = g|_{V_{x}^{0}} (and x ∈ X and y ∈ Y(x) are uniquely determined by these conditions). We have x = ∏_{i∈I} g_{i} and y = ∏_{i∉I} g_{i}.

We show that these definitions allow us to define (X , Y)-sections.

Proposition 4.3. Take any integer d > 0, and let X = X_{d}, X_{(d)}, X_{d}^{(s)} or X_{(d)}^{(s)}. Then, for any x ∈ X,

(i) Y(x) is a union of conjugacy classes of C_{G}(x). (ii) For all y ∈ Y(x), C_{G}(xy) ≤ C_{G}(x). (iii) For all g ∈ G, Y(x^{g}) = Y(x)^{g}. (iv) Two elements of xY(x) are G-conjugate if and only if they are C_{G}(x)- conjugate.

(v) G = ∐_{x∈X /G}{(xy)^{G}, y ∈ Y(x)/C_{G}(x)}.

Proof. (i) We have C_{G}(x) = H_{x} × H_{x}^{0}, and we see C_{G}(x) as a subgroup of GL(V_{x}) × GL(V_{x}^{0}). For y ∈ Y(x), y = (y|_{V_{x}}, y|_{V_{x}^{0}}) = (1, y_{x}^{0}). Then, for all h = (h_{x}, h_{x}^{0}) ∈ C_{G}(x), h^{−1}yh = (1, (h_{x}^{0})^{−1}y_{x}^{0}h_{x}^{0}) ∈ Y(x). (ii) For all y ∈ Y(x), we have C_{G}(xy) ≅ C_{GL(V_{x})}(x|_{V_{x}}) × C_{GL(V_{x}^{0})}(y|_{V_{x}^{0}}) ≤ C_{GL(V_{x})}(x|_{V_{x}}) × GL(V_{x}^{0}) ≅ C_{G}(x), and C_{G}(xy) ≤ C_{G}(x) since the isomorphism on the left is the restriction to C_{GL(V_{x})}(x|_{V_{x}}) × C_{GL(V_{x}^{0})}(y|_{V_{x}^{0}}) of the isomorphism on the right. (iii) Take g ∈ G and y ∈ Y(x). Then x^{g} ∈ X and F_{x^{g}} = F_{x} (and thus F_{x^{g}}^{0} = F_{x}^{0}). We have V = g^{−1}V = g^{−1}V_{x}⊕g^{−1}V_{x}^{0}. Furthermore, x^{g}|_{g^{−1}V_{x}^{0}} = 1, g^{−1}V_{x} is x^{g}-stable, and x^{g}|_{g^{−1}V_{x}} ∈ (...f^{ν(f)}...)_{f∈F_{x}}. Thus g^{−1}V_{x} = V_{x^{g}} and g^{−1}V_{x}^{0} = V_{x^{g}}^{0}. Now, since y ∈ Y(x), we see that y^{g}|_{g^{−1}V_{x}} = 1, g^{−1}V_{x}^{0} is y^{g}- stable, and y^{g}|_{g^{−1}V_{x}^{0}} ∈ (...f^{ν(f)}...)_{f∈F_{x}^{0}}. Hence y^{g} ∈ Y(x^{g}), and Y(x)^{g} ⊂ Y(x^{g}) for all g ∈ G. Then, for any g ∈ G, Y(x^{g})^{g^{−1}} ⊂ Y(x) so that Y(x^{g}) ⊂ Y(x)^{g}. Hence the result. (iv) Suppose that, for some y, z ∈ Y(x), there exists h ∈ G such that xy = h^{−1}xzh. Writing g = xy, we also have g = x^{h}z^{h}, and x^{h} ∈ X (since X is a union of G-conjugacy classes) and z^{h} ∈ Y(x^{h}) (by (iii)). By the uniqueness of such a writing for g, we have x^{h} = x and z^{h} = y. Hence h ∈ C_{G}(x). In particular, xy and xz are C_{G}(x)-conjugate.

(v) For any g ∈ G, there exists a unique x ∈ X such that g ∈ xY(x). Thus

G = ∐_{x∈X} xY(x)

= ∐_{x∈X /G}(xY(x))^{G} (because of (iii))

= ∐_{x∈X /G} ⋃_{y∈Y(x)}(xy)^{G}

= ∐_{x∈X /G} ∐_{y∈Y(x)/C_{G}(x)}(xy)^{G},

this last equality being true by (iv) (if (xy)^{g} = (xz)^{g′}, then xy = xz^{g′g^{−1}} so that, by (iv), y and z are C_{G}(x)-conjugate).

This proposition shows that the (X , Y)-sections we defined correspond to the notion of (X , Y)-sections presented in [18].

For any x ∈ X, we call the union of the G-conjugacy classes meeting xY(x) the Y-section of x. We remark that the Y-sections of G are quite different from ordinary prime sections. Take any 1 ≠ x ∈ X. Then, by definition, Y(x) ⊂ Y(1)∩C_{G}(x). However, if d ≠ 1, then there exists λ ∈ F_{q}^{×} such that λI_{n} ∈ Y(1) ∩ C_{G}(x) but λI_{n} ∉ Y(x), so that Y(x) ≠ Y(1) ∩ C_{G}(x) (while this equality holds when we define (X , Y)-sections to be the ordinary ℓ-sections for some prime ℓ).

Furthermore, still supposing d ≠ 1, if x is a (non trivial) ℓ-element of G for some prime ℓ, then, most of the time (that is, when q − 1 is not a power of ℓ), there exists an ℓ-regular element λ ∈ F_{q}^{×} such that λI_{n} ∉ Y(x), so that xλI_{n} ∉ (xY(x))^{G}. But xλI_{n} belongs to the ℓ-section of x. Hence the Y-section of x is not a union of ℓ-sections.

4.3 (X , Y)-blocks

4.3.1 Blocks

Take any integer d > 0, and let X = X_{d}, X_{(d)}, X_{d}^{(s)} or X_{(d)}^{(s)}. We define on Irr(G) the relation ∼ of direct Y(1)-linking: for χ, ψ ∈Irr(G), χ ∼ ψ if and only if

< χ, ψ >_{Y(1)}= \frac{1}{|G|} ∑_{y∈Y(1)} χ(y)\overline{ψ(y)} ≠ 0.

Extending ∼ by transitivity, we obtain an equivalence relation ≈ on Irr(G). We define an (X , Y)-block of G to be an equivalence class of the equivalence relation ≈. We will also consider the restriction of ≈ to the subset Unip(G) of unipotent characters of G. The equivalence classes of Unip(G) will be called unipotent (X , Y)-blocks of G. It is clear that, for any (X , Y)-block B of G, B ∩ Unip(G) is a union of unipotent (X , Y)-blocks.

4.3.2 Irreducible characters of G

For the results we give here, we refer to [12].

4.3. (X , Y)-BLOCKS 115

The unipotent characters of G = GL(n, q) are the irreducible components of the permutation representation of G on the cosets of a Borel subgroup (i.e. the normalizer in G of a Sylow p-subgroup of G, where p is the defining characteristic). They are labelled by the partitions of n. For λ ⊢ n, we will write χ_{λ} the unipotent (irreducible) character of G labelled by λ.

Each irreducible character of G can be parametrized as χ_{s,μ}, where s is a semisimple element of G and χ_{μ} is a unipotent character of C_{G}(s) (note that, since s is semisimple, C_{G}(s) is a product of linear groups GL(k, q^{δ}) where δ is the degree of an irreducible factor of Char(s) and k is its multiplicity in Char(s); in particular, μ may be seen as a sequence of partitions (μ_{1}, …, μ_{t}) (of k_{1}, …, k_{t} respectively), and χ_{μ} = χ_{μ_{1}} ⊗ ⋯ ⊗ χ_{μ_{t}}). We obtain a complete minimal set of representatives by taking representatives (s, μ) for the Gconjugacy classes of such pairs. The set of characters χ_{s,μ}, for μ varying, is the geometric conjugacy class s_{G} of s.

More precisely, if χ ∈Irr(G), then there exists s ∈ G semisimple, there exists ψ, unipotent character of H = C_{G}(s), and there exist signs ε_{G} and ε_{H} such that

χ = ε_{G}ε_{H}R_{H}^{G}(ŝψ),

where R_{H}^{G} is the additive operator, defined in the Deligne-Lusztig theory, from X(H) to X(G) (character rings of representations of H and G respectively over \overline{Q_{ℓ}}, an algebraic closure of the ℓ-adic field Q_{ℓ}), and ŝ is the linear character of the center Z(H) of H given by:

∀φ ∈ s_{H}, ∀h ∈ H, ∀t ∈ Z(H), φ(th) = ŝ(t)φ(h).

If the unipotent character ψ is labelled by the tuple of partitions μ, we write ψ = χ_{μ}, and χ = χ_{s,μ} = ±R_{H}^{G}(ŝχ_{μ}). We introduce two class functions χ^{μ} and χ^{s,μ} (of H and G respectively) such that χ_{μ} = ±χ^{μ}, and χ^{s,μ} = R_{H}^{G}(ŝχ^{μ}) = ∓χ_{s,μ}.

4.3.3 Murnaghan-Nakayama Rule for unipotent characters

Pick g ∈ G, and write Char(g) = ∏_{i} f_{i}^{k_{i}} and the corresponding decomposition g = ∏_{i} g_{i}. Then pick i_{0}, and write g = ρσ, where

ρ = g_{i_{0}} and σ = ∏_{i≠i_{0}} g_{i}.

Writing d the degree of f_{i_{0}}, m = k_{i_{0}}d, and l = n − m, we have, writing ∽ for equivalence of matrices over F_{q},

g ∽ ( \begin{matrix}. & . & . \\ & & & U_{ν_{i}}(f_{i}) \\ & & & & . & . & . \end{matrix} ) ,

ρ ∽ ( \begin{matrix}I_{l} \\ & U_{ν_{i_{0}}}(f_{i_{0}}) \end{matrix} )

(and we may consider that ρ ∈ G_{0} = GL(m, q)),

σ ∽ ( \begin{matrix}() \\ & . & . & . \\ & & & & () \\ & & & & & I_{m} \end{matrix} )

(and we may consider that σ ∈ G_{1} = GL(l, q)).

Then, using the results on the Jordan decomposition given in the first section, we see that the semi-simple part ρ_{S} of ρ is equivalent to

( \begin{matrix}I_{l} \\ & U_{k_{i_{0}}}(f_{i_{0}}) \end{matrix} ) .

We have ρ_{S} ∈ GL(n, q) and Char(ρ_{S}) = f_{i_{0}}^{k_{i_{0}}}(X − 1)^{l}, and we may consider that ρ_{S} ∈ G_{0} = GL(m, q) and Char(ρ_{S}) = f_{i_{0}}^{k_{i_{0}}}.

We have C_{G}(ρ_{S}) = H = H_{0} × H_{1}, where H_{1} ≅ GL(l, q) and H_{0} = C_{GL(m,q)}(ρ_{S}) ≅ GL(k_{i_{0}}, q^{d}).

Theorem 4.4 (Murnaghan-Nakayama Rule). (cf [12])

Let g ∈ G, and ρ and σ be as above, and let ν ⊢ n. Then

χ^{ν}(ρσ) = ∑_{λ∈L_{ν}} a_{νλ}^{ρ}χ^{λ}(σ),

where L_{ν} is the set of partitions λ of l which can be obtained from ν by removing k_{i_{0}} d-hooks, and a_{νλ}^{ρ} ∈ Z[q^{d}]. If L_{ν} = ∅, then χ^{ν}(ρσ) = 0. We have a_{νλ}^{ρ} ≠ 0 for λ ∈ L_{ν}. (The coefficients of a_{νλ}^{ρ} depend on the characters of the symmetric group S_{k_{i_{0}}} and the Green functions of GL(k_{i_{0}}, q^{d}) ≅ H_{0} (applied to the unipotent part ρ_{U} of ρ), and all the non-zero coefficients of a_{νλ}^{ρ} have the same sign).

Remark: it is easy to see that, if a_{νλ}^{ρ} ≠ 0, then ν and λ have the same d-core.

The idea is to use this theorem recursively so as to be able to obtain information about < χ^{λ}, χ^{μ} >_{xY(x)}, for λ, μ ⊢ n and x ∈ X_{d} or x ∈ X_{(d)}. We first use it to obtain a formula for χ^{μ}(xy), where μ ⊢ n and y ∈ Y(x). We let X = X_{d} or X_{(}d), and take 1 ≠ x ∈ X. Suppose x ∈ c_{x} = (… f_{i}^{λ_{i}} …) where, for each i, λ_{i} ⊢ k_{i}, and δ(f_{i}) = d (resp. d|δ(f_{i})) or f_{i} = X − 1. If f_{i} ≠ X − 1, we let δ(f_{i}) = m_{i}d. From the definition of X, we see that, in the primary decomposition of x, we may omit the component corresponding to X − 1, because it is necessarily the identity. We therefore write x = x_{1} … x_{r} where each x_{i} has exatly one elementary divisor distinct from X − 1, and with the same multiplicity as in Char(x). We will say that x has d-type (k_{1}m_{1}, …, k_{r}m_{r}).

By repeated use of the Murnaghan-Nakayama Rule, we obtain that, for any y ∈ Y(x) and for μ ⊢ n,

χ^{μ}(xy) = ∑_{μ_{1}∈L_{μ}} a_{μμ_{1}}^{x_{1}}χ^{μ_{1}}(x_{2} … x_{r}y)
= ∑_{μ_{1}∈L_{μ}} a_{μμ_{1}}^{x_{1}} (∑_{μ_{2}∈L_{μ_{1}}} a_{μ_{1}μ_{2}}^{x_{2}}χ^{μ_{2}}(x_{3} … x_{r}y))
= ∑_{μ_{1}∈L_{μ}} … ∑_{μ_{r}∈L_{μ_{r−1}}} a_{μμ_{1}}^{x_{1}} … a_{μ_{r−1}μ_{r}}^{x_{r}}χ^{μ_{r}}(y)

which can be written

χ^{μ}(xy) = ∑_{λ∈L_{(k_{1}m_{1},...,krmr)d}^{μ}} α_{μλ}^{(x_{1},...,x_{r})}χ^{λ}(y),

where the α_{μλ}^{(x_{1},...,x_{r})}’s are integers and L_{(k_{1}m_{1},...,k_{r}m_{r})d}^{μ} is the set of partitions of n−(∑_{i} k_{i}m_{i})d which can be obtained from μ by removing k_{1} m_{1}d-hooks, then k_{2} m_{2}d-hooks, …, and finally k_{r} m_{r}d-hooks. We will call such a sequence of removals a (k_{1}m_{1}, …, k_{r}m_{r})d-path from μ to λ.

Note that, in this sum, each λ can appear several times, if there is more than one (k_{1}m_{1}, …, k_{r}m_{r})d-path from μ to λ. Note also that, in the right side of this equality, y has implicitely been seen as an element of GL(l, q), where l = n − (∑_{i} k_{i}m_{i})d.

If L_{(k_{1}m_{1},...,k_{r}m_{r})d}^{μ} = ∅, then χ^{μ}(xy) = 0. We have α_{μλ}^{(x_{1},...,x_{r})} ∈ Z[q^{d}], and, if we separate the possibly multiple occurences of each λ in the sum, then, for λ ∈ L_{(k_{1}m_{1},...,k_{r}m_{r})d}^{μ}, each of the α_{μλ}^{(x_{1},...,x_{r})}’s is non-zero (these are indexed by the (k_{1}m_{1}, …, k_{r}m_{r})d-paths from μ to λ).

If α_{μλ}^{(x_{1},...,x_{r})} ≠ 0, then, since there is a (k_{1}m_{1}, …, k_{r}m_{r})d-path from μ to λ, and since the removal of a hook of length md can be obtained by the removal of a sequence of m hooks of length d, we see that μ and λ have the same d-core.

We call the α_{μλ}^{(x_{1},...,x_{r})}’s the MN-coefficients, and we will from now on write α_{μλ}^{x} for α_{μλ}^{(x_{1},...,x_{r})}.

4.3.4 Nakayama Conjecture for unipotent blocks

Now, we are able to prove that the unipotent characters of G satisfy one sense of a generalized Nakayama Conjecture. The proof we give is an adaptation to our case of the proof given by Külshammer, Olsson and Robinson in the case of symmetric groups. We let X = X_{d} or X_{(d)} and take x ∈ X of dtype km = (k_{1}m_{1}, …, k_{r}m_{r}). We let l = n − kmd = l − (∑_{i=1}^{r} k_{i}m_{i})d. Writing G_{0} = GL(kmd, q) = ∏_{i=1}^{r} GL(k_{i}m_{i}d, q) and G_{1} = GL(l, q), we have x = (x_{0}, x_{1}) = (x_{0}, 1) ∈ G_{0} × G_{1}. Then C_{G}(x) = H_{0} × H_{1}, where H_{1} ≅ GL(l, q) and H_{0} = C_{G_{0}}(x_{0}) (note that, if x is semi-simple, then H_{0} ≅ GL(k, q^{md}) ≅ ∏_{i=1}^{r} GL(k_{i}, q^{m_{i}d})).

Now take y ∈ Y(x). Then, as an element of C_{G}(x) = H_{0} × H_{1}, we have y = (y_{0}, y_{1}) = (1, y_{1}).

Writing Y_{t}(u) for Y(u) when u ∈ GL(t, q), we have that y, element of C_{G}(x), belongs to Y_{n}(x) if and only if y = (1, y_{1}) ∈ H_{0} ×H_{1} where y_{1} belongs to Y_{l}(1). Hence Y_{n}(x) is in natural one to one correspondence with Y_{l}(1). Now we consider μ, μ′ ⊢ n, and x ∈ X of d-type km = (k_{1}m_{1}, …, k_{r}m_{r}). We

have

< χ^{μ}, χ^{μ′} >_{xY(x)} = \frac{1}{|G|} ∑_{y∈Y(x)} χ^{μ}(xy)\overline{χ^{μ′}(xy)}

= \frac{1}{|G|} ∑_{y=(y_{0},y_{1})∈Y(x)} [( ∑_{λ∈L_{km}^{μ}} α_{μλ}^{x}χ^{λ}(y_{1})) ( ∑_{λ′∈L_{km}^{μ′}} α_{μ′λ′}^{x}\overline{χ^{λ′}(y_{1})})]

= \frac{1}{|G|} ∑_{y_{1}∈Y_{l}(1)} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x}χ^{λ}(y_{1})\overline{χ^{λ′}(y_{1})}

(by the above remark on Y(x)).

We write A_{μμ′}^{x} =< χ^{μ}, χ^{μ′} >_{xY(x)}. Then

A_{μμ′}^{x} = \frac{1}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x} ∑_{y_{1}∈Y_{l}(1)} χ^{λ}(y_{1})\overline{χ^{λ′}(y_{1})}

= \frac{|H_{1}|}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x} < χ^{λ}χ^{λ′} >_{Y_{l}(1)}

i.e. A_{μμ′}^{x} = \frac{|H_{1}|}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x}A_{λλ′}^{1}.

We use induction on n to prove that, if A_{μμ′}^{x} ≠ 0, then μ and μ′ have the same d-core. We may assume that μ ≠ μ′.

If n < d, then each partition is its own d-core. Furthermore, in this case, X = {1} and Y(1) = G. Thus, for all x ∈ X, we have A_{μμ′}^{x} = A_{μμ′}^{1}, and A_{μμ′}^{1} =< χ^{μ}, χ^{μ′} >_{G}= 0 (since μ ≠ μ′). Hence the result is true in this case. Thus, we suppose n ≥ d. First suppose x ≠ 1 and A_{μμ′}^{x} ≠ 0. Then

A_{μμ′}^{x} = \frac{|H_{1}|}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x}A_{λλ′}^{1} ≠ 0.

Thus there exist λ ∈ L_{km}^{μ} and λ′ ∈ L_{km}^{μ′} such that α_{μλ}^{x}α_{μ′λ′}^{x}A_{λλ′}^{1} ≠ 0. Then α_{μλ}^{x} ≠ 0 implies that μ and λ have the same d-core, and α_{μ′λ′}^{x} ≠ 0 implies that μ′ and λ′ have the same d-core. And, by the induction hypothesis (applied to n − kmd < n), A_{λλ′}^{1} ≠ 0 implies that λ and λ′ have the same d-core.

Now, if x = 1, we see, by the existence and uniqueness of the decomposition we introduced, that

0 =< χ^{μ}, χ^{μ′} >_{G}= ∑_{x∈X} < χ^{μ}, χ^{μ′} >_{xY(x)}= ∑_{x∈X} A_{μμ′}^{x}.

Hence, if A_{μμ′}^{1} ≠ 0, then there exists an x′ ∈ X \{1} such that A_{μμ′}^{x′} ≠ 0. This in turn implies, by the previous case, that μ and μ′ have the same d-core. Skipping back from class functions to irreducible characters, we see that we have proved the following

Theorem 4.5. If two unipotent (irreducible) characters χ_{μ} and χ_{μ′} of G = GL(n, q) are directly linked across some xY(x), where x ∈ X = X_{d} or X_{(d)}, then μ and μ′ have the same d-core (and this is true in particular for x = 1).

Extending by transitivity the relation of direct Y(1)-linking, we obtain

Theorem 4.6. Let X = X_{d} or X_{(d)}. If two unipotent characters χ_{μ} and χ_{μ′} of G = GL(n, q) are in the same unipotent (X , Y)-block of G, then μ and μ′ have the same d-core.

Each unipotent (X , Y)-block of G is therefore associated to a d-core. For each given d-core γ, we can consider the union of the (possibly many) unipotent (X , Y)-blocks associated to γ. The (a priori) bigger blocks obtained in this way are parametrized by the set of d-cores of partitions of n, and they satisfy the following equivalent of the Nakayama Conjecture. In accordance with the terminology used in [18], we call them combinatorial unipotent (X , Y)-blocks.

Theorem/Definition 4.7. Let X = X_{d} or X_{(d)}. Two unipotent (irreducible) characters χ_{μ} and χ_{μ′} belong to the same combinatorial unipotent (X , Y)- block of GL(n, q) if and only if μ and μ′ have the same d-core.

We can find in [12] an analogue of the Murnaghan-Nakayama rule which applies to any irreducible character of G, with the extra hypothesis that the element ρ is semisimple and primary, with an elementary divisor of degree d = 1. It is also mentioned that this theorem can be generalized to an arbitrary d. This should allow us to derive the same properties as above for (X , Y)-blocks of the whole of Irr(G), but taking this time X = X_{d}^{(s)} or X_{(d)}^{(s)}. The (X , Y)- blocks of G obtained in this way should satisfy one sense of a generalized Nakayama Conjecture, and we should be able to define combinatorial (X , Y)- blocks of G satisfying this generalized Nakayama Conjecture. Note that the properties we proved for unipotent characters hold if we take X = X_{d}^{(s)} or X_{(d)}^{(s)}.

In any case, it remains to study the possible difference between (X , Y)- blocks and combinatorial (X , Y)-blocks.

4.4 Second Main Theorem property

We now want to show that the combinatorial unipotent (X , Y)-blocks of G satisfy the Second Main Theorem property. We therefore need to define, for x ∈ X, the (X , Y)-blocks of C_{G}(x), and then to look at their behaviour with respect to domination.

4.4.1 (X , Y)-blocks of centralizers

We take any positive integer d, and X = X_{d}, X_{(d)}, X_{d}^{(s)} or X_{(d)}^{(s)}. Even though we don’t know what they look like, this defines (X , Y)-blocks of Irr(G) (which are just Y(1)-blocks, built using Y(1)-linking), and we can define (X , Y)-blocks of C_{G}(x) for each x ∈ X.

For x ∈ X \ {1}, the (X , Y)-blocks of C_{G}(x) are the equivalence classes of the transitive closure of direct xY(x)-linking. Equivalently, irreducible characters in distinct (X , Y)-blocks of C_{G}(x) are orthogonal across xY(x), and the blocks are minimal for this property. If x has d-type km, then, writing l = n − kmd, we have C_{G}(x) = H_{0} × H_{1} where H_{1} ≅ GL(l, q) and H_{0} ≤ G_{0} ≅ GL(kmd, q). Then Irr(C_{G}(x)) =Irr(H_{0})⊗Irr(H_{1}). Note that, as we noted in the previous section, we may consider x as an element of H_{0}, and then xY(x) = {(x, y) ∈ H_{0} × H_{1}, y ∈ Y_{l}(1)}.

Take χ_{0}, ψ_{0} ∈ Irr(H_{0}) and χ_{1}, ψ_{1} ∈ Irr(H_{1}). We have

< χ_{0} ⊗ χ_{1}, ψ_{0} ⊗ ψ_{1} >_{xY(x)} = \frac{1}{|C_{G}(x)|} ∑_{y∈Y(x)}(χ_{0} ⊗ χ_{1})(xy)\overline{(ψ_{0} ⊗ ψ_{1})(xy)}

= \frac{1}{|C_{G}(x)|} ∑_{y∈Y_{l}(1)} χ_{0}(x)χ_{1}(y)\overline{ψ_{0}(x)}\overline{ψ_{1}(y)}

= \frac{χ_{0}(x)\overline{ψ_{0}(x)}}{|C_{G}(x)|} ∑_{y∈Y_{l}(1)} χ_{1}(y)\overline{ψ_{1}(y)}

= \frac{χ_{0}(x)\overline{ψ_{0}(x)}}{|H_{0}|} < χ_{1}, ψ_{1} >_{Y_{l}(1)} .

Since x is central in H_{0}, we have χ_{0}(x)\overline{ψ_{0}(x)} ≠ 0, and we see that χ_{0} ⊗ χ_{1} and ψ_{0} ⊗ ψ_{1} are directly xY(x)-linked if and only if χ_{1} and ψ_{1} are directly Y_{l}(1)-linked. Extending by transitivity, we obtain that the (X , Y)-blocks of C_{G}(x) are the Irr(H_{0}) ⊗ b_{i}’s, where b_{i} runs through the Y_{l}(1)-blocks of H_{1} ≅ GL(l, q).

In analogy with this, we define the unipotent (X , Y)-blocks and combinatorial unipotent (X , Y)-blocks of C_{G}(x) to be the Irr(H_{0})⊗b_{i}’s, where b_{i} runs through the unipotent Y_{l}(1)-blocks and combinatorial unipotent Y_{l}(1)-blocks of H_{1} ≅ GL(l, q) respectively.

4.4.2 Second Main Theorem property for combinatorial unipotent (X , Y)-blocks

We take any positive integer d, X = X_{d} or X_{(d)}, and x ∈ X \ {1}. We write C_{G}(x) = H_{0} × H_{1} as above. For any combinatorial unipotent (X , Y)-block B of G, labelled by the d-core γ, we set β(x, B) = Irr(H_{0}) ⊗ b, where b is the Y_{l}(1)-block of H_{1} labelled by γ. For any χ_{μ} ∈ B and ψ_{0} ⊗ ψ_{λ} ∈ Irr(H_{0}) ⊗ b, we set

c_{χ_{μ},ψ_{0}⊗ψ_{λ}} = \{ \begin{matrix}α_{μλ}^{x} if ψ_{0} = 1_{H_{0}} \\ 0 otherwise \end{matrix} ,

where the α_{μλ}^{x}’s are the MN-coefficients, obtained from the Murnaghan-Nakayama rule for unipotent characters. Then the definition of the α_{μλ}^{x}’s shows that, for each x ∈ X, the β(x, B)’s and c_{χ_{μ},ψ_{0}⊗ψ_{λ}}’s satisfy the hypotheses of Proposition 1.33. Indeed, for each combinatorial unipotent (X , Y)- block B, for each χ_{μ} ∈ B and for each y ∈ Y(x), we have

χ(xy) = ∑_{ψ_{0}⊗ψ_{λ}∈β(x,B)} c_{χ_{μ},ψ_{0}⊗ψ_{λ}}ψ_{0} ⊗ ψ_{λ}(xy),

and, furthermore, β(x, B) and β(x, B′) are disjoint whenever B and B′ are distinct combinatorial unipotent (X , Y)-blocks of G. This implies that, for each x ∈ X, each combinatorial unipotent (X , Y)-block of C_{G}(x) is dominated by a unique combinatorial unipotent (X , Y)-block of G. Hence the combinatorial unipotent (X , Y)-blocks of G satisfy the Second Main Theorem property.

(...), which just shows that the human brain is ill-adapted for thinking and was probably originally designed for cooling the blood.

Terry Pratchett, The last hero.

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