Jean-Baptiste GRAMAIN Université Cl. Bernard, Lyon 1 DEA de Mathématiques 2001-2002

PERFECT ISOMETRIES

Mémoire de DEA

Tuteur :

Meinolf GECK

Introduction

The representation theory of finite groups appeared one century ago. We will suppose the reader to be familiar with its ideas and techniques, especially with the notion of group characters and the structures of character rings and character tables.

This mmoire has a double purpose. In a first part, we shortly describe the modular representation theory (introduced by R. Brauer), corresponding to a partition of the irreducible characters of a group according to blocks, and therefore also referred to as Block Theory.

The study of an important invariant of this theory, namely the defect group of a block lead M. Brou to conjecture the existence of a strong relationship between the character rings of a block and a block of the normalizer of its defect group (in case this is Abelian). This relationship is illustated by the concept of Perfect Isometry, presented in Part 2, the second objective of this mmoire being to study a precise perfect isometry, in an ”easy” case of Brou’s Conjecture (Part 3).

Contents

1 BLOCK THEORY 4

1.1 Context, notations . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Semi-simplicity, absolute simplicity . . . . . . . . . . . 4 1.1.2 p-modular systems . . . . . . . . . . . . . . . . . . . . 5

1.2 Brauer characters . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Blocks and idempotents . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Blocks of G . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Representations of Z(KG) and Z(kG) . . . . . . . . . 9 1.3.3 Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Cartan matrix, elementary divisors . . . . . . . . . . . . . . . 10 1.4.1 The Cartan matrix . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Elementary divisors . . . . . . . . . . . . . . . . . . . . 11

1.5 Defect groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 The Brauer correspondence . . . . . . . . . . . . . . . . . . . 13

2 PERFECT ISOMETRIES 15 2.1 Generalised induction . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Perfect characters . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Virtual characters vanishing on p-singular elements . . . . . . 18 2.4 Structure of ZIrr_{p−sing}(KG) . . . . . . . . . . . . . . . . . . . 18 2.5 Blocks, perfect isometries . . . . . . . . . . . . . . . . . . . . . 20 2.6 Class functions . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 An algebra isomorphism . . . . . . . . . . . . . . . . . . . . . 24 2.8 Back to blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 BROUÉ’S CONJECTURE 30

3.1 The symmetric group S_{p} . . . . . . . . . . . . . . . . . . . . . 31 3.1.1 Context, notations . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Characters of the symmetric group . . . . . . . . . . . 31 3.1.3 Blocks of defect one . . . . . . . . . . . . . . . . . . . . 33 3.1.4 Principal block of G = S_{p} . . . . . . . . . . . . . . . . 33 3.1.5 Structure of N = N_{G}(P) . . . . . . . . . . . . . . . . . 35 3.1.6 Ordinary irreducible characters of N . . . . . . . . . . 37 3.1.7 Principal block . . . . . . . . . . . . . . . . . . . . . . 38

3.2 A perfect isometry . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 A virtual character . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Proposition 28 (ii) . . . . . . . . . . . . . . . . . . . . 41 3.2.4 Conjugacy classes of N and G . . . . . . . . . . . . . . 41 3.2.5 Values on p-elements . . . . . . . . . . . . . . . . . . . 42 3.2.6 Back to 3.2.4 . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.7 The solution . . . . . . . . . . . . . . . . . . . . . . . . 46

Part 1

BLOCK THEORY

In this part, we introduce the modular representation theory of finite groups. For all the results we will mention, we refer to [6].

1.1 Context, notations

1.1.1 Semi-simplicity, absolute simplicity

In all this work, we let G be a finite group. Then, 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. 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 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 precizes 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 representation of \overline{F}_{p}G can be realised over k_{0}.

1.1.2 p-modular systems

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 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 = (p). 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 Meinolf Geck’s lectures on modular representation theory, and, for another presentation, we refer to M.F. Atiyah and I.G. Macdonald (Introduction to commutative algebra, Addison-Wesley, 1969).

In the sequel, we will note z ⟼ z̄ the canonical surjection O ⟶ O/M, also called reduction modulo p..

Definition : 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.

for G K (and thus taking simple kG-module

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 f.d. \overline{F}_{p}G-module can be realised over k.

Definition : 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 absolutey simple.

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 3 Let U_{p′} = {x ∈ K^{×} | x^{n} = 1 for some n ≥ 1, n| |G|, 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′} ⟶ {x̄ ∈ k^{×} | x̄^{m} = 1} ⊂ k^{×}

is a bijection.

Now if M is a kG-module, associated to the representation ρ of dimension d, and g ∈ G is p-regular of order ∘(g), then the eigenvalues (ξ_{i})_{1≤i≤d} of ρ(g) are ∘(g)-th roots of unity, so are in k^{×}. Therefore, there are unique w_{i} ∈ U_{p′} such that \overline{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} .

It is constant on p-regular conjugacy classes, and, for all g ∈ G_{p′}, \overline{φ_{M}(g)} is the character of M evaluated at g. Moreover, if M ≅_{kG} M′, then φ_{M} = φ_{M′}.

We have the following

Theorem 4 Let

Irr_{K}(G) = {V_{1}, …, V_{n}} and χ_{i} = χ_{V_{i}}
Irr_{k}(G) = {M_{1}, …, M_{r}} and φ_{i} = φ_{M_{i}}.

(i) The functions φ_{i}: G_{p′} ⟶ K are linearly independant over K. (ii) For all i, 1 ≤ i ≤ n, there exist unique (d_{ij})_{1≤j≤r} ∈ N^{r} such that

χ_{i}(g) = ∑_{j=1}^{r} d_{ij}φ_{j}(g) ∀g ∈ G_{p′}.

The d_{ij}’s are the p-modular decomposition numbers of G, and D = ((d_{ij}))_{1≤i≤n,1≤j≤r} is the decomposition matrix of G. We will write χ̂_{i} for the restriction χ_{i}|_{G_{p′}}. From the above, we deduce

Proposition 5 (i) The rank of D is r ≤ n. (ii) |Irr_{k}(G)| = r = |{p-regular conjugacy classes of G}| (iii) If g_{1}, …, g_{r} ∈ G are representatives for the p-regular classes, then

det (((φ_{i}(g_{j})))_{1≤i,j≤r}) ∈ O^{×}.

(iv) If p |̸ |G|, then r = n and {χ̂_{1}, …, χ̂_{n}} = {φ_{1}, …, φ_{r}}. If p | |G|, then r < n.

Remark : note that (iv) implies that, if p |̸ |G| and (K, O, k) is a p-modular system for G such that any finite dimensional CG-module can be realised over K (and thus over O), then the elements of Irr(kG) are obtained by taking the reduction modulo p of the elements of Irr(KG), and any simple kG-module is absolutely simple.

We let IBr_{p}(G) = {φ_{i}, 1 ≤ i ≤ r}. For each Brauer character φ ∈ IBr_{p}(G), we define the class function

Φ_{φ} = ∑_{χ∈Irr(KG)} d_{χφ}χ.

These functions will turn out to be great tools in the sequel. They satisfy :

Proposition 6 (i) If g ∈ G is p-singular, then Φ_{φ}(g) = 0. (ii) If g ∈ G is p-regular, then Φ_{φ}(g)/|C_{G}(g)| ∈ O.

In particular, |G|_{p} | Φ_{φ}(1). (iii) For all φ, φ′ ∈ IBr_{p}(G),

\frac{1}{|G|} ∑_{g∈G_{p′}} Φ_{φ}(g^{−1})φ′(g) = δ_{φφ′}.

1.3 Blocks and idempotents

1.3.1 Blocks of G

Wedderburn’s Theorem gives us

KG ≅ ⨁_{i=1}^{n} M_{d_{i}}(K)

where Irr(KG) = {χ_{1}, …, χ_{n}}, and χ_{i} is the character of ρ_{i}: KG ⟶ M_{d_{i}}(K). Via this isomorphism, we can transport the structure of Z(⨁_{i=1}^{n} M_{d_{i}}(K)) = K.Id_{d_{1}} ⨁ ⋯ ⨁ K.Id_{d_{n}} to obtain a similar decomposition into blocks for KG. Namely :

Proposition 7 For 1 ≤ i ≤ n, let

e_{i} = e_{χ_{i}} = \frac{χ_{i}(1)}{|G|} ∑_{g∈G} χ_{i}(g^{−1})g ∈ KG.

Then e_{i}e_{j} = δ_{ij}e_{i}, 1 = e_{1} + ⋯ + e_{n}, e_{i} ∈ Z(KG), and {e_{1}, … e_{n}} is a basis for Z(KG).

We have KG = ⨁_{χ∈Irr(KG)} KGe_{χ} (direct sum of ideals). The KGe_{χ}’s are called the blocks of KG.

We will be interested in p-blocks, i.e. ideals OGe of OG, where e ∈ Z(OG) is a primitive idempotent. One can see that any idempotent in Z(OG) must be of the form ∑_{χ∈A} e_{χ} for some A ⊂ Irr(KG). We define the Brauer Graph Γ(G, p) : the vertices are the χ ∈ Irr(KG), and χ, χ′ ∈ Irr(KG) (χ ≠ χ′) are linked if there exists φ ∈ IBr_{p}(G) such that d_{χφ} ≠ 0 ≠ d_{χ′φ}. The connected components of Γ(G, p) correspond to a decomposition of D into blocks.

Theorem 8 Let A ⊂ Irr(KG). Then e_{A} = ∑_{χ∈A} e_{χ} is a primitive idempotent of Z(OG) if and only if A is the set of characters in a connected component of Γ(G, p).

By a slight abuse of terminology, we see elements of Irr(KG) and IBr_{p}(G) as belonging themselves to blocks of G. We will denote by Bl(G) the set of blocks of G. Irr(KG) and IBr_{p}(G) can be reordered so that the decomposition matrix D is diagonal by blocks. Each of these blocks will be called decomposition matrix for the corresponding block of G.

For B a block, we let e_{B} = ∑_{χ∈B} e_{χ}, and, for each class C of G, we let

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

where C̃ is the sum of the elements of C and g_{C} ∈ C. Then e_{B} = ∑_{C} a_{B}(C̃)C̃, and a_{B}(C̃) = 0 if C consists in p-singular elements.

1.3.2 Representations of Z(KG) and Z(kG)

To each χ ∈ Irr(KG) is associated an irreducible representation ω_{χ} of Z(KG) given by

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 of C.

For χ, χ′ ∈ Irr(KG), we have ω_{χ}(e_{χ′}) = δ_{χχ′}. Moreover, for all χ ∈ Irr(KG), \overline{ω_{χ}} is an irreducible representation of Z(kG). More precisely

Proposition 9 For B a block, \overline{ω_{χ}} is independant of the choice of χ ∈ B. If we let ω_{B} = \overline{ω_{χ}} for some χ ∈ B, then \{ω_{B}, B ∈ Bl(G)\} is the complete set of distinct irreducible representations of Z(kG), \{\overline{e_{B}}, B ∈ Bl(G)\} is the complete set of primitive idempotents of Z(kG), and ω_{B}(\overline{e_{B′}}) = δ_{BB′} for all B, B′ ∈ Bl(G).

1.3.3 Defect

Definition : For χ ∈ Irr(KG), define the defect of χ, d(χ), to be the exact power of p dividing the integer |G|/χ(1), and define the defect of a block B to be

d(B) = max_{χ∈B} d(χ).

Finally, for χ ∈ B, define the height of χ to be h(χ) = d(B) − d(χ).

For B a block and φ in B, we would like to have a relation between φ(1) and |G|, as we have one for ordinary irreducible characters. We have Proposition 10 If B is a block of G and |G|_{p} = p^{a}, then p^{a−d(B)} | φ(1) for all φ ∈ B, and p^{a−d(B)+1} |̸ φ(1) for some φ ∈ B.

The case of blocks of defect zero is explained by

Proposition 11 Let B be a block of G and χ ∈ B. Then the following are equivalent :

(i) d(B) = 0. (ii) d(χ) = 0. (iii) χ is the only irreducible character in B. (iv) χ vanishes on p-singular elements.

1.4 Cartan matrix, elementary divisors

1.4.1 The Cartan matrix

Theorem 12 (Brauer’s reciprocity)

Let φ ∈ IBr_{p}(G). Then there exists a primitive idempotent e_{φ} ∈ OG such that the character of the OG-module OGe_{φ} is given by

Φ_{φ} = ∑_{χ∈Irr(KG)} d_{χφ}χ.

If IBr_{p}(G) = {φ_{1}, …, φ_{r}}, then for each i, there exists a primitive idempotent ε_{i} ∈ kG such that

\overline{OGe_{φ_{i}}} = kGε_{i} =: P_{i}.

The Brauer character of P_{i} is Φ̂_{φ_{i}}.

Remark : The P_{i}’s are representatives for the projective indecomposable kG-modules. If φ_{i} is the Brauer character of M_{i} and rad(P_{i}) is the radical of P_{i}, then P_{i}/rad(P_{i}) ≅ M_{i}. Moreover,

P_{i} ≅ P_{j} ⟺ M_{i} ≅ M_{j}.

Definition : For any 1 ≤ i, j ≤ r, let c_{ij} be the multiplicity of M_{i} in a composition series for P_{j}. Then C = ((c_{ij}))_{1≤i,j≤r} is called the Cartan matrix of kG.

We then have Theorem 13 (Brauer)

C = D^{t}D

where D is the decomposition matrix. (In particular, C is symmetric)

Consequently, C has the same block structure as D. We suppose now that Irr(KG) and IBr_{p}(G) have been reordered once and for all so that C and D are in block form. For a block B of G, we define the Cartan matrix C_{B} of B as we defined its decomposition matrix D_{B}. For any block B ∈ Bl(G), we thus have C_{B} = D_{B}^{t}D_{B}.

1.4.2 Elementary divisors

Remember that D ∈ M_{n,r}(Z), where n = |Irr(KG)| and r = |IBr_{p}(G)|.

Proposition 14 There exist U ∈ GL_{n}(Z) and V ∈ GL_{r}(Z) such that

UDV = ( \begin{matrix}Id_{r} \\ (0) \end{matrix} )

For B ∈ Bl(G), we note r_{B} (resp. n_{B}) the number of irreducible Brauer characters (resp. ordinary irreducible characters) in the block B.

Then D_{B} ∈ M_{n_{B},r_{B}}(Z) and there exist U_{B} ∈ GL_{n_{B}}(Z) and V_{B} ∈ GL_{r_{B}}(Z) such that

U_{B}D_{B}V_{B} = ( \begin{matrix}Id_{r_{B}} \\ (0) \end{matrix} )

Theorem 15 Let B be a block of G. Then

(i) The elementary divisors of 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)}. (iii) If χ ∈ B and d(χ) = 1, then d(B) = 1.

Noting X the set of p-regular columns of the ordinary character table of G and Φ the Brauer character table, we have X = DΦ. Now the orthogonality relations yield

X^{t}X = diag(|C_{G}(g_{1})|, … |C_{G}(g_{r})|)

where g_{1}, …, g_{r} are representatives for the p-regular classes of G.

For any class C, we define the defect of C, d(C) = |C_{G}(g)| for g ∈ C. Then the theory of elementary divisors in O ⊃ Z shows Theorem 16 The elementary divisors of the Cartan matrix C are

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

Remark : The comparison between these two theorems yields

Proposition 17 Let P ∈ Syl_{p}(G). Then the number of blocks of G of maximal defect is equal to the number of p-regular classes of N_{G}(P) contained in C_{G}(P).

1.5 Defect groups

As blocks of full defect are connected to Sylow p-subgroups of G, we will see that each block of positive defect is connected to a canonically determined conjugacy class of p-groups.

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 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).

Proposition-Definition 18 If B is any block of G, then there exists a pregular class C such that ω_{B}(C̃) ≠ 0 ≠ \overline{a_{B}}(C̃). Such a class is called a defect class for B, and δ(C) is called a defect group for B.

Proposition 19 Let δ(B) be a defect group for the block B and C a class of G. Then

(i) If ω_{B}(C̃) ≠ 0, then δ(B) ⊂_{G} δ(C). (ii) If \overline{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 20 If P is a normal p-subgroup of G and B is a block of G, then P ⊂_{G} δ(B).

1.6 The Brauer correspondence

Let H be a subgroup of G. If λ is any linear functional on Z(kH) and C is a 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 21 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 C a class 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 22 The map μ: Z(kG) ⟶ Z(kH) is an algebra homomorphism, called the Brauer homomorphism.

Theorem 23 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(\overline{e_{B}}) = ∑_{b^{G}=B} \overline{e_{b}}.

The principal result is

Theorem 24 (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.

Finally, let’s say a word about principal blocks. The principal block is the one containing the trivial character. It is therefore of full defect, so has for defect group a Sylow p-subgroup. Furthermore,

Proposition 25 Suppose H is a subgroup of G, b is a block of H and C_{G}(δ(b)) ⊂ H. Then b^{G} 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 of N_{G}(P).

The following two parts are meant to define and manipulate a deeper kind of correspondence, this one at the level of characters, namely perfect isometries. Once again, the principal blocks of G and N_{G}(P) will turn out to be deeply connected.

PERFECT ISOMETRIES

This part is devoted to the introduction of the notion of perfect isometry, and the establishment of its first properties, presented in [1] and [2].

We consider G and H two finite groups, p a prime and (K, O, k) a pmodular system large enough for both G and H.

2.1 Generalised induction

Let μ be a virtual character of G × H, i.e. μ ∈ ZIrr(K(G × H)). We define on ZIrr(KH) a linear application I_{μ} by letting, for ζ ∈ Irr(KH) and g ∈ G,

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

Then

I_{μ}: ZIrr(KH) ⟶ ZIrr(KG)

Proof : It suffices to show that, if ζ ∈ Irr(KH), then I_{μ}(ζ) ∈ ZIrr(KG).

(i) I_{μ}(ζ) is a class function : for all u, g ∈ G

\begin{aligned}I_{μ}(ζ)(ugu^{−1}) & = & \frac{1}{|H|} ∑_{h∈H} μ(ugu^{−1}, h^{−1})ζ(h) \\ & = & \frac{1}{|H|} ∑_{h∈H} μ(g, h^{−1})ζ(h) = I_{μ}(ζ)(g) \end{aligned}

(ii) For all χ ∈ Irr(KG), < I_{μ}(ζ), χ >∈ Z, where < , > denotes the scalar product on the space of class functions on G :

\begin{aligned}< I_{μ}(ζ), χ > & = & \frac{1}{|G|} ∑_{g∈G} I_{μ}(ζ)(g)χ(g^{−1}) \\ & = & \frac{1}{|H|} ∑_{h∈H} ζ(h)\frac{1}{|G|} ∑_{g∈G} μ(g, h^{−1})χ(g^{−1}) \end{aligned}

We can write

\begin{aligned}μ = ∑_{i,j} n_{ij}χ_{i} ⊗ ψ_{j} & , n_{ij} ∈ Z & , χ_{i} ∈ Irr(KG) \\ & & , ψ_{i} ∈ Irr(KH) \end{aligned}

Thus

\frac{1}{|G|} ∑_{g∈G} μ(g, h^{−1})χ(g^{−1}) = \frac{1}{|G|} ∑_{g∈G} ∑_{i,j} n_{ij}χ_{i}(g)ψ_{j}(h^{−1})χ(g^{−1})
= ∑_{i,j} n_{ij}ψ_{j}(h^{−1}) \underbrace{\frac{1}{|G|} ∑_{g∈G} χ_{i}(g)χ(g^{−1})}
= q_{i} ∈ Z

and

< I_{μ}(ζ), χ >= ∑_{i,j} n_{ij}q_{i} \underbrace{\frac{1}{|H|} ∑_{h∈H} ζ(h)ψ_{j}(h^{−1})} ∈ Z.

∈ Z

This ends the proof.

I_{μ} is called generalised induction. Similarly, we define the generalised restriction

R_{μ}: ZIrr(KG) ⟶ ZIrr(KH)

by letting, for χ ∈ Irr(KG) and h ∈ H

R_{μ}(χ)(h) = \frac{1}{|G|} ∑_{g∈G} μ(g^{−1}, h)χ(g).

Then I_{μ} and R_{μ} are adjoint to each other, i.e. for all χ ∈ Irr(KG), ζ ∈ Irr(KH),

< R_{μ}(χ), ζ >_{H}=< χ, I_{μ}(ζ) >_{G}

Indeed

< R_{μ}(χ), ζ >_{H}= \frac{1}{|G||H|} ∑_{h∈H} ∑_{g∈G} μ(g^{−1}, h^{−1})χ(g)ζ(h) =< χ, I_{μ}(ζ) >_{G}

We will use the following

Proposition 26 Any homomorphism J: ZIrr(KH) ⟶ ZIrr(KG) is of the form J = I_{μ} for some virtual character μ of G × H.

Proof : If Irr(KH) = {ψ_{1}, …, ψ_{m}} and Irr(KG) = {χ_{1}, …, χ_{n}}, and if

J(ψ_{j}) = ∑_{i=1}^{n} a_{ij}χ_{i} for all 1 ≤ j ≤ m

(a_{ij} ∈ Z), then, taking

μ = ∑_{i, j} a_{ij}χ_{i} ⊗ ψ_{j} ∈ ZIrr(K(G × H))

in the computation page 15, we get, for all 1 ≤ i_{0} ≤ n, 1 ≤ j_{0} ≤ m,

\begin{aligned}< I_{μ}(ψ_{j_{0}}), χ_{i_{0}} > & = & ∑_{i, j} a_{ij} (\frac{1}{|G|} ∑_{g∈G} χ_{i}(g)χ_{i_{0}}(g^{−1})) (\frac{1}{|H|} ∑_{h∈H} ψ_{j_{0}}(h)ψ_{j}(h^{−1})) \\ & = & ∑_{i, j} a_{ij}δ_{ii_{0}}δ_{jj_{0}} = a_{i_{0}j_{0}} \end{aligned}

Hence J = I_{μ}.

2.2 Perfect characters

Definition : μ, virtual character of G × H, is said to be perfect if (per. 1) For all g ∈ G, 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 p-singular.

Remark : By construction of O (localization of A_{K} in P), (per. 1) is equivalent to :

for all g ∈ G, h ∈ H, \frac{μ(g, h)}{|C_{G}(g)|} ∈ O and \frac{μ(g, h)}{|C_{H}(h)|} ∈ O.

Exemple : If H is a subgroup of G, and μ is the character of the (KG, KH)- module KG, then μ is perfect, I_{μ} = Ind_{H}^{G} and R_{μ} = Res_{H}^{G} are the classical induction and restriction.

2.3 Virtual characters vanishing on p-singular elements

Let ζ ∈ ZIrr(KH). Suppose ζ(h) = 0 for each p-singular h ∈ H, and μ is a perfect character of G × H.

Then, if g ∈ G and I_{μ}(ζ)(g) ≠ 0, since

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

there exists h ∈ H such that μ(g, h^{−1})ζ(h) ≠ 0. Then ζ(h) ≠ 0 implies that h is p-regular, and μ(g, h^{−1}) ≠ 0 implies (via (per. 2)) that g is also p-regular.

Therefore I_{μ}(ζ) vanishes on p-singular elements of G. If we denote by ZIrr_{p−sing}(KH) the set of virtual characters of KH which vanish on the p-singular elements, we get, by restriction,

I_{μ}: ZIrr_{p−sing}(KH) ⟶ ZIrr_{p−sing}(KG).

2.4 Structure of ZIrr_{p−sing}(KG)

We note Irr(KG) = {χ_{i} , 1 ≤ i ≤ n} and IBr_{p}(G) = {φ_{i} , 1 ≤ i ≤ r}. D = ((d_{χφ})) the decomposition matrix of G. For φ ∈ IBr_{p}(G), we have

Φ_{φ} = ∑_{χ∈Irr(KG)} d_{χφ}χ.

Then, by Proposition 6 (i), Φ_{φ} ∈ ZIrr_{p−sing}(KG). Moreover, the rank of D is the number r = IBr_{p}(G) of p-regular classes in G. The (Φ_{φ})_{φ∈IBr_{p}(G)}, since corresponding to the columns of D, are linearly independant over Q.

We will show that the (Φ_{φ})_{φ∈IBr_{p}(G)} form a basis for the Z-module ZIrr_{p−sing}(KG).

We have

ZIrr_{p−sing}(KG) = {∑_{i=1}^{n} n_{i}χ_{i}; ∑_{i=1}^{n} n_{i}χ_{i}(g) = 0 ∀g ∈ G p-singular}

As dim_{Z}(ZIrr(KG)) = |Irr(KG)| = number of p-regular classes + number of p-singular classes, we have

dim_{Z}(ZIrr_{p−sing}(KG)) = |{p-regular classes}|
= dim_{Z}(Z [(Φ_{φ})_{φ∈IBr_{p}(G)}]).

This doesn’t show that the Φ_{φ}’s generate the Z-module ZIrr_{p−sing}(KG) (only that the Z-submodule they generate has the same dimension). However, it shows that, for all ζ ∈ ZIrr_{p−sing}(KG), we can write

ζ = ∑_{φ∈IBr_{p}(G)} m_{ζφ}Φ_{φ} with m_{ζφ} ∈ Q.

Now there exist P ∈ GL_{n}(Z) and Q ∈ GL_{r}(Z) such that

P DQ = ( \begin{matrix}Id_{r} \\ (0) \end{matrix} ) = Δ.

Noting

X = ( χ̂_{1} ; ⋮ ; χ̂_{n} ) and Φ = ( φ_{1} ; ⋮ ; φ_{r} )

we have X = DΦ = P^{−1}ΔQ^{−1}Φ.

Hence P X = ΔQ^{−1}Φ, MP X = Q^{−1}Φ, and Φ = QMP X, where M = (Id_{r} (0)) ∈ M_{r,n}(Z). Thus

∀φ ∈ IBr_{p}(G), φ = ∑_{χ∈Irr(KG)} a_{φχ}χ̂ with a_{φχ} ∈ Z.

But then, if we denote by < , >_{p′} the scalar prduct in ZIrr_{p−sing}(KG), for all ζ ∈ ZIrr_{p−sing}(KG), φ′ ∈ IBr_{p}(G)

< ζ, φ′ >_{p′}= ∑_{φ∈IBr_{p}(G)} m_{ζφ} < Φ_{φ}, φ′ >_{p′}= ∑_{φ∈IBr_{p}(G)} m_{ζφ}δ_{φφ′} = m_{ζφ′}

and

< ζ, φ′ >_{p′}= ∑_{χ∈Irr(KG)} a_{φχ} < ζ, χ̂ >_{p′}∈ Z.

Therefore, the (Φ_{φ})_{φ∈IBr_{p}(G)} form a basis for the Z-module ZIrr_{p−sing}(KG).

2.5 Blocks, perfect isometries

The decomposition matrices of G and H have decompositions by blocks, corresponding to the decompositions of the algebras OG and OH into direct sums of blocks of type OGe and OHf, where e and f are central idempotents of OG and OH respectively. We take two blocks A = OGe and B = OHf, where e and f are central idempotents (not necessarily primitive idempotents). Thus A and B may be sums of blocks as defined in Part 1.

Definition : If μ is a perfect character of G × H, we say that I_{μ} is a perfect isometry between A and B if I_{μ} induces a bijective isometry between ZIrr(KB) and ZIrr(KA).

Proposition 27 (first properties) Suppose that A and B are perfectly isometric via I_{μ}. We denote by D_{A} and D_{B} the decomposition matrices of KA and KB respectively.

(i) D_{A} and D_{B} have the same number of lines (the common dimension of ZIrr(KB) and ZIrr(KA)). (ii) For all χ ∈ Irr(KB), I_{μ}(χ) ∈ ZIrr(KA), thus, since I_{μ} is an isometry,

< I_{μ}(χ), I_{μ}(χ) >_{G}=< χ, χ >_{H}= 1

so that I_{μ}(χ) = ±ψ, for some ψ ∈ Irr(KA). (iii) I_{μ} maps each column of D_{B} to a linear combination with integer coefficients of the columns of D_{A} (up to reordering the lines of D_{A}, and changing signs, according to the order of the lines of D_{B} and to the relations given by (ii) ). (iv) D_{A} and D_{B} have the same number of columns.

Proof : only (iv) requires proof.

We have, by restriction,

I_{μ}: ZIrr_{p−sing}(KB) ⟶ ZIrr_{p−sing}(KA).

Thus dim_{Z}(ZIrr_{p−sing}(KB)) = |IBr_{p}(B)| = |{ columns of D_{B} }| is less than or equal to dim_{Z}(ZIrr_{p−sing}(KA)) = |{ columns of D_{A} }|.

Similarly, by restriction,

R_{μ}: ZIrr_{p−sing}(KA) ⟶ ZIrr_{p−sing}(KB).

Therefore the number of columns of D_{A} is less than or equal to the number of columns of D_{B}.

I_{μ} being a bijective isometry, I_{μ}(ZIrr_{p−sing}(KB)) is a Z-submodule of ZIrr_{p−sing}(KA) of same dimension. In fact, one can prove better using property (iii).

We have D_{A}, D_{B} ∈ M_{n,r}(Z), rk(D_{A}) = rk(D_{B}) = r. The matrix D_{B} contains the scalar products

< χ, Φ_{φ} >_{H} , χ ∈ Irr(KB) , φ ∈ IBr_{p}(B).

Property (iii) can then be reinterpreted as :

if we denote by U ∈ M_{n,n}(Z) the matrix corresponding to the permutation of the lines of D_{A} and to their occasional change of sign, and by V ∈ M_{r,r}(Z) the matrix corresponding to the linear combinations of the columns of D_{A}, then UD_{A}V contains the scalar products

< I_{μ}(χ), I_{μ}(Φ_{φ}) >_{G} , χ ∈ Irr(KB) , φ ∈ IBr_{p}(B).

And, as I_{μ} is an isometry, we finally obtain

D_{B} = UD_{A}V.

Applying the same arguments to R_{μ}, we obtain U′ ∈ M_{r,r}(Z), V′ ∈ M_{n,n}(Z) such that D_{A} = U′D_{B}V′.

But, R_{μ} and I_{μ} being adjoint to each other, for all χ ∈ Irr(KB), ψ ∈ Irr(KA)

< R_{μ}(ψ), χ >_{H}=< ψ, I_{μ}(χ) >_{G}

thus

I_{μ}(χ) = ±ψ ⟹ R_{μ}(ψ) = ±χ

and thus U′ = U^{−1} (and I_{μ}^{−1} = R_{μ}).

Moreover, by definition of U, U^{−1} = U^{t} (U is ”almost” a permutation matrix). Thus

D_{B} = UD_{A}V = UU^{−1}D_{B}V′V = D_{B}V′V

and

D_{A} = U^{−1}D_{B}V′ = U^{−1}UD_{A}V V′ = D_{A}V V′.

Now V, V′ ∈ M_{r,r}(Z) and rk(D_{A}) = rk(D_{B}) = r, so D_{A} (resp. D_{B}) contains a submatrix I_{A} ∈ GL_{r}(Z) (resp. I_{B} ∈ GL_{r}(Z) and we have

I_{A} = I_{A}V V′ , I_{B} = I_{B}V′V.

Thus V′ = V^{−1} ∈ GL_{r}(Z). Finally,

D_{B} = UD_{A}V , U ∈ GL_{n}(Z) , V ∈ GL_{r}(Z).

The invertibility of V yields

I_{μ}(ZIrr_{p−sing}(KB)) = ZIrr_{p−sing}(KA).

Considering then the Cartan matrices C_{A} and C_{B}, we obtain

C_{B} = D_{B}^{t}D_{B} = V^{t}D_{A}^{t}U^{t}UD_{A}V = V^{t}D_{A}^{t}D_{A}V = V^{t}C_{A}V.

In particular, C_{B} and C_{A} are equivalent over Z.

2.6 Class functions

For e an idempotent in Z(OG), we denote by CF (G, e, K) the K-vector space of central functions α on G, with values in K, and such that α(eg) = α(g) for all g ∈ G, and by CF_{p′}(G, e, K) the subspace of functions which vanish on p-singular elements of G. We define in the same way CF (G, e, O) and CF_{p′}(G, e, O). Irr(KG, e) is a K-basis for CF (G, e, K). {Φ_{φ} , φ ∈ IBr_{p}(G, e)} is a K-basis for CF_{p′}(G, e, K).

We have

Proposition 28 The character μ is perfect if and only if

(i) I_{μ} maps CF (H, f, O) into CF (G, e, O) and R_{μ} maps CF (G, e, O) into CF (H, f, O). (ii) I_{μ} maps CF_{p′}(H, f, K) into CF_{p′}(G, e, K) and R_{μ} maps CF_{p′}(G, e, K) into CF_{p′}(H, f, K).

Proof : Suppose first that μ is a perfect character of G×H. The assertions (i) and (ii) are only reinterpretations of properties (per. 1) and (per. 2) of μ: (ii) is shown as in section 3

If ζ ∈ CF_{p′}(H, f, K), then I_{μ}(ζ) ∈ CF (G, e, K) and

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

thus, via (per. 2),

I_{μ}(ζ)(g) ≠ 0 ⟹ ∃h ∈ H such that μ(g, h^{−1})ζ(h) ≠ 0
⟹ g is p-regular

therefore I_{μ}(ζ) ∈ CF_{p′}(G, e, K). Similarly for R_{μ}.

As for (i), it uses (per. 1). We consider h_{1}, …, h_{n} representatives for the conjugacy classes of H. If ζ ∈ CF (H, f, O), then I_{μ}(ζ) ∈ CF (G, e, K) and

∀g ∈ G , I_{μ}(ζ)(g) = ∑_{i=1}^{n} \frac{1}{|C_{H}(h_{i})|}μ(g^{−1}, h_{i})ζ(h_{i}) ∈ O

by (per. 1), thus I_{μ}(ζ) ∈ CF (G, e, O). Similarly for R_{μ}.

Conversely, suppose (i) and (ii) are true. Consider h_{1}, … h_{n} representatives for the conjugacy classes in H, and define the (α_{i})_{1≤i≤n} on KH by

α_{i}(h) = 1_{Cl_{H}(h_{i})f}(hf).

The α_{i}’s belong to CF (H, f, O).

Let g ∈ G. For all i, 1 ≤ i ≤ n

I_{μ}(α_{i})(g) = \frac{1}{|H|} ∑_{j=1}^{n} μ(g, h_{j})α_{i}(h_{j}^{−1}) = \frac{μ(g, h_{i})}{|C_{H}(h_{i})|} ∈ O

by (ii). Thus

∀g ∈ G , ∀h ∈ H , \frac{μ(g, h)}{|C_{H}(h)|} ∈ O.

Analogously (with the β_{i}(.) = 1_{Cl_{G}(g_{i})e}(.e) ∈ CF (G, e, O) and R_{μ}), we obtain

∀g ∈ G , ∀h ∈ H , \frac{μ(g, h)}{|C_{G}(g)|} ∈ O.

Thus μ satisfies (per. 1).

Remark that α_{i} ∈ CF_{p′}(H, f, K) if and only if h_{i} is p-regular. Suppose (g, h) ∈ G × H is such that μ(g, h) ≠ 0.

Then, if h is p-regular, so is h^{−1}, thus h^{−1} ∈ Cl_{H}(h_{i}) for some p-regular h_{i}, and α_{i} ∈ CF_{p′}(H, f, K). Hence I_{μ}(α_{i}) ∈ CF_{p′}(G, e, K) and

I_{μ}(α_{i})(g) = \frac{1}{|C_{H}(h_{i})|}μ(g, h) ≠ 0

therefore g is p-regular.

Analogously, if g is p-regular, R_{μ}(β_{j})(h) ≠ 0 for some β_{j} ∈ CF_{p′}(G, e, K), so h is p-regular.

Thus, μ satisfies (per. 2) and, therefore, μ is perfect.

2.7 An algebra isomorphism

Suppose

I_{μ}: ZIrr(KH, f) ⟶ ZIrr(KG, e)

is a perfect isometry, and

R_{μ}: ZIrr(KG, e) ⟶ ZIrr(KH, f)

is its adjoint.

If ζ ∈ Irr(KH, f)

I_{μ}(ζ) = ±χ , χ ∈ Irr(KG, e).

We let e_{H}(ζ) (resp. e_{G}(χ)) be the primitive central idempotent associated to ζ (resp. χ).

e_{H}(ζ) = \frac{ζ(1)}{|H|} ∑_{h∈H} ζ(h^{−1})h ∈ Z(KH)f

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

We have seen

Theorem 29 I_{μ} induces a bijection between the primitive idempotents of ZKHf and the primitive idempotents of ZKGe.

We will show that this bijection induces an algebra isomorphism between ZOHf and ZOGe. We let

\begin{aligned}I_{μ}^{0} : & ZKHf & ⟶ & ZKGe \\ & y = ∑_{h∈H} y(h)h & ⟼ & I_{μ}^{0}(y) = ∑_{g∈G} (\frac{1}{|H|} ∑_{h∈H} μ(g^{−1}, h)y(h)) g \end{aligned}

and

\begin{aligned}R_{μ}^{0} : & ZKGe & ⟶ & ZKHf \\ & z = ∑_{g∈G} z(g)g & ⟼ & R_{μ}^{0}(z) = ∑_{h∈H} (\frac{1}{|G|} ∑_{g∈G} μ(g, h^{−1})z(g)) h \end{aligned}

NB : y = ∑_{h∈H} y(h)h ∈ KH is central if and only if h ⟼ y(h) is a class function on H.

Supposing furthermore that y ∈ ZOHf, i.e. that y(h) ∈ O for all h ∈ H, we have, taking h_{1}, …, h_{n} representatives for the conjugacy classes in H and Cl_{H}(h_{i}) the class of h_{i},

\frac{1}{|H|} ∑_{h∈H} μ(g^{−1}, h)y(h) = ∑_{i=1}^{n} \frac{|Cl_{H}(h_{i})|}{|H|}μ(g^{−1}, h_{i})y(h_{i})
= ∑_{i=1}^{n} \underbrace{\frac{1}{|C_{H}(h_{i})|}μ(g^{−1}, h_{i})} \underbrace{y(h_{i})} ∈ O
∈ O by (per. 1) ∈ O

Thus I_{μ}^{0}(y) ∈ ZOGe.

Similarly, since for all g ∈ G, h ∈ H, \frac{μ(g,h^{−1})}{|C_{G}(g)|} ∈ O,

∀z ∈ ZOHe , R_{μ}^{0}(z) ∈ ZOHf.

Therefore

I_{μ}^{0}|_{ZOHf}: ZOHf ⟶ ZOGe

and

R_{μ}^{0}|_{ZOGe}: ZOGe ⟶ ZOHf.

Furthermore, the K-linear (additive) morphisms I_{μ}^{0} and R_{μ}^{0}, when restricted to ZOHf and ZOGe respectively, are O-linear.

Finally, I_{μ}^{0}|_{ZOHf} and R_{μ}^{0}|_{ZOGe} are (additive) isomorphisms, inverses to one another. As a matter of fact : {e_{H}(ζ) , ζ ∈ Irr(KH, f)} is a basis for

ZKHf, and {e_{G}(χ) , χ ∈ Irr(KG, e)} is a basis for ZKGe.

Now if ζ ∈ Irr(KH, f), I_{μ}(ζ) = ±χ = ε(ζ)χ, for some χ ∈ Irr(KG, e). Then

< I_{μ}(ζ), χ >_{G}=< ζ, R_{μ}(χ) >_{H}= ε(ζ)

so that, since R_{μ}(χ) ∈ ±Irr(KH, f), R_{μ}(χ) = ε(ζ)ζ. Therefore, if I (bijection between primitive idempotents induced by I_{μ}) maps e_{H}(ζ) to e_{G}(χ), then R (bijection between primitive idempotents induced by R_{μ}) maps e_{G}(χ) to e_{H}(ζ).

We compute

\begin{aligned}I_{μ}^{0}(e_{H}(ζ)) & = & ∑_{g∈G} (\frac{1}{|H|} ∑_{h∈H} μ(g^{−1}, h)\frac{ζ(1)}{|H|}ζ(h^{−1})) g \\ & = & \frac{ζ(1)}{|H|} ∑_{g∈G} (\underbrace{\frac{1}{|H|} ∑_{h∈H} μ(g^{−1}, h)ζ(h^{−1})}) g \\ & & = I_{μ}(ζ)(g^{−1}) \\ & = & \frac{ζ(1)}{|H|}ε(ζ) ∑_{g∈G} χ(g^{−1})g \end{aligned} .

Thus

I_{μ}^{0}(e_{H}(ζ)) = \frac{ζ(1)}{|H|} \frac{|G|}{χ(1)}e_{G}(χ)

and, similarly,

R_{μ}^{0}(e_{G}(χ)) = \frac{χ(1)}{|G|} \frac{|H|}{ζ(1)}e_{H}(ζ).

Therefore

R_{μ}^{0}(I_{μ}^{0}(e_{H}(ζ))) = e_{H}(ζ) , I_{μ}^{0}(R_{μ}^{0}(e_{G}(χ))) = e_{G}(χ)

and

R_{μ}^{0} ∘ I_{μ}^{0} = Id_{ZKHf} , I_{μ}^{0} ∘ R_{μ}^{0} = Id_{ZKGe}.

Hence R_{μ}^{0} and I_{μ}^{0} are mutual inverse isomorphisms between the K-algebras ZKHf and ZKGe. By restriction to O, R_{μ}^{0} and I_{μ}^{0} therefore induce Olinear isomorphisms inverses to one another between ZOHf and ZOGe.

To obtain our algebra isomorphism, it remains to deal with the property of multiplicative morphism, which R_{μ}^{0} and I_{μ}^{0} don’t have. We define

\begin{aligned}Ĩ_{μ}: & ZKHf & ⟶ & ZKGe \\ & y & ⟼ & I_{μ}^{0}(yR_{μ}^{0}(e)) \end{aligned}

Then Ĩ_{μ} linearly extends to ZKHf the bijection I (induced by I_{μ} between primitive idempotents of ZKHf and ZKGe).

Proof :

R_{μ}^{0}(e) = R_{μ}^{0} ( ∑_{χ∈Irr(KG,e)} e_{G}(χ))
= ∑_{χ∈Irr(KG,e)} R_{μ}^{0}(e_{G}(χ))

= ∑_{ψ∈Irr(KH,f)} \frac{|H|/ψ(1)}{|G|/χ(1)}e_{H}(ψ) with R(e_{G}(χ)) = e_{H}(ψ)

Thus, for all ζ ∈ Irr(KH, f)

Ĩ_{μ}(e_{H}(ζ)) = I_{μ}^{0}(e_{H}(ζ)R_{μ}^{0}(e)) = I_{μ}^{0} (\frac{|H|/ζ(1)}{|G|/χ(1)}e_{H}(ζ))

where χ is such that R(e_{G}(χ)) = e_{H}(ζ), and thus such that I(e_{H}(ζ)) = e_{G}(χ). Hence

Ĩ_{μ}(e_{H}(ζ)) = \frac{|H|/ζ(1)}{|G|/χ(1)} \frac{|G|/χ(1)}{|H|/ζ(1)}e_{G}(χ) = e_{G}(χ)

which ends the proof.

In particular, Ĩ_{μ} is an algebra homomorphism. As a matter of fact, on the basis (e_{H}(ζ))_{ζ∈Irr(KH,f)}, we have : for all ζ, ζ′ ∈ Irr(KH, f), if χ and χ′ are such that I(e_{H}(ζ)) = e_{G}(χ) and I(e_{H}(ζ′)) = e_{G}(χ′)

\begin{aligned}Ĩ_{μ}(e_{H}(ζ)e_{H}(ζ′)) & = & I_{μ}^{0}(e_{H}(ζ)e_{H}(ζ′)R_{μ}^{0}(e)) \\ & = & \{ \begin{matrix}I_{μ}^{0}(0) = 0 & if ζ ≠ ζ′ \\ I_{μ}^{0}(e_{H}(ζ)R_{μ}^{0}(e)) = e_{G}(χ) & if ζ = ζ′ \end{matrix} \end{aligned}

and

Ĩ_{μ}(e_{H}(ζ))Ĩ_{μ}(e_{H}(ζ′)) = e_{G}(χ)e_{G}(χ′) = \{ \begin{matrix}0 & if ζ ≠ ζ′ \\ e_{G}(χ) & if ζ = ζ′ \end{matrix}

which proves our assertion.

Thus Ĩ_{μ} is an algebra isomorphism, with inverse

\begin{aligned}R̃_{μ}: & ZKGe & ⟶ & ZKHf \\ & z & ⟼ & R_{μ}^{0}(zI_{μ}^{0}(f)) \end{aligned}

Finally, since e ∈ ZOGe and f ∈ ZOHf, and since I_{μ}^{0} and R_{μ}^{0} are ”defined on O”, Ĩ_{μ} and R̃_{μ} induce isomorphisms, inverses to one another, between ZOHf and ZOGe. We have therefore proved :

Theorem 30 The bijection between primitive idempotents of ZKHf and ZKGe defined by I_{μ} induces an algebra isomorphism between ZOHf and ZOGe.

2.8 Back to blocks

The fact to have now an algebra isomorphism between ZOHf and ZOGe allows us to go back to the blocks of these algebras. Indeed, Ĩ_{μ}, being an algebra isomorphism, maps any idempotent of ZOHf to an idempotent of ZOGe, and any primitive idempotent of ZOHf to a primitive idempotent of ZOGe.

To say it differently, I_{μ} induces a bijection between the blocks of H associated to f and the blocks of G associated to e. Using the same arguments as at the beginning of this Part, we see that this bijection preserves the number of ordinary and Brauer irreducible characters, as well as the elementary divisors of the Cartan matrix. We will show that it also preserves the defect and height of ordinary irreducible characters.

Lemma 31 If ζ ∈ Irr(KH, f), and if χ_{ζ} = ε(ζ)I_{μ}(ζ), then \frac{|G|/χ_{ζ}(1)}{|H|/ζ(1)} is invertible in O.

Proof : We know that \frac{|G|/χ_{ζ}(1)}{|H|/ζ(1)} ∈ Q. Moreover,

I_{μ}^{0}(f) = ∑_{ζ∈Irr(KH,f)} \frac{|G|/χ_{ζ}(1)}{|H|/ζ(1)}e_{G}(χ_{ζ}) ∈ ZOGe

and

\begin{aligned}R_{μ}^{0}(e) & = & ∑_{χ∈Irr(KG,e)} \frac{|H|/ζ_{χ}(1)}{|G|/χ(1)}e_{H}(ζ_{χ}) \\ & = & ∑_{ζ∈Irr(KH,f)} \frac{|H|/ζ(1)}{|G|/χ_{ζ}(1)}e_{H}(ζ) ∈ ZOHf \end{aligned}

(this last equality being true because R_{μ} and I_{μ} are mutual inverses).

Therefore

\frac{|G|/χ_{ζ}(1)}{|H|/ζ(1)} ∈ O and \frac{|H|/ζ(1)}{|G|/χ_{ζ}(1)} ∈ O

for all ζ ∈ Irr(KH, f). Hence the result.

In particular,

∀ζ ∈ Irr(KH, f) , (|G|/χ_{ζ}(1))_{p} = (|H|/ζ(1))_{p} (†)

The defects d(e) and d(f) of e and f are the integers defined by

p^{d(e)} = lcm{(|G|/χ(1))_{p} , χ ∈ Irr(KG, e)}
p^{d(f)} = lcm{(|H|/ζ(1))_{p} , ζ ∈ Irr(KH, f)}

and the height of ζ ∈ Irr(KH, f) is the integer h(ζ) such that (|H|/ζ(1))_{p} = p^{d(f)−h(ζ)}.

It is then clear with (†) that d(f) = d(e) and that, for all ζ ∈ Irr(KH, f),

p^{d(f)−h(ζ)} = (|H|/ζ(1))_{p} = (|G|/χ_{ζ}(1))_{p} = p^{d(e)−h(χ_{ζ})}

so that h(ε(ζ)I_{μ}(ζ)) = h(χ_{ζ}) = h(ζ), as announced.

BROUÉ’S CONJECTURE

Now that we have in hands (almost) all the tools we need, we can state the following

Conjecture 32 (Broué) Let G be a finite group whose Sylow p-subgroups are abelian. Let H be the normalizer of one of the Sylow p-subgroups of G. Then the principal blocks B_{0}(G) and B_{0}(H) of G and H are perfectly isometric.

This conjecture is known to be true in the following cases :

• for all p, if G is p-solvable;

• for p = 2, in all cases;

• for all p, if G is a sporadic simple group;

• for all p, if G is a symmetric group or an alternating group;

• if G is the group of rational points of a connected reductive algebraic group G defined over F_{q} and p is a prime number which does not divide q and which is good for G.

Conjecture 32 is in fact a weaker version of others conjectures, which, under the same hypotheses, state respectively that there is a normalized isotypy between B_{0}(G) and B_{0}(H), and that B_{0}(G) and B_{0}(H) are Rickard equivalent. For these notions and further references for proofs in special cases, we refer to [2] and [3].

In the sequel, we will remain at the level of perfect isometries, and study a very special case, where G is the symmetric group S_{p}.

3.1 The symmetric group S_{p}

3.1.1 Context, notations

In all the sequel, we let p be a prime and G = S_{p}. We let N = N_{G}(P) be the normalizer of some Sylow p-subgroup P of G. Since |G|_{p} = p, we have |P | = p, so that |N|_{p} = p and P =< τ > for some p-cycle τ.

The principal blocks B_{0} of G and b_{0} of N, being of maximal defect, have thus defect 1.

Finally, we denote by (K, O, k) a p-modular system, large enough for both G and N.

Our aim is to show that B_{0} and b_{0} are perfectly isometric. In this section, we will give descriptions of G and N, B_{0}, b_{0} and their decomposition matrices. The construction and analysis of the perfect isometry will be postponed to section 3.2.

Remark : In most cases, our figures will be drawn for p odd. However, the reasoning applies also to p = 2.

(which

of

χ has

3.1.2 Characters of the symmetric group

We let n be a fixed positive integer. Then the conjugacy classes of the symmetric group S_{n} are parametrized by the partitions of n (corresponding to the decompositions in product of disjoint cycles). In particular, every element of S_{n} is conjugated to its inverse, so that the complex irreducible characters of S_{n} are real-valued. In fact, we even have :

Theorem 33 The ordinary character table of S_{n} has integer entries.

For any partition α of n, we write α = (a_{1}, …, a_{r}) with a_{1}, …, a_{r} ∈ N, a_{1} ≥ ⋯ ≥ a_{r} > 0, and a_{1} + ⋯ + a_{r} = n.

We then define the Young diagram [α] assciated to α to be the set of points (i, j) ∈ (Z^{+})^{2} satisfying 1 ≤ i ≤ r and, for each such i, 1 ≤ j ≤ a_{i}.

As an example, for α = 13^{2}4, [α] will be written

We define the hook length of a box in a Young diagram to be the number of squares directly below or directly to the right of the box, including the box once. Labelling each box by its hook length in the above diagram, we get

1457 235 124 1

Given a partition α of n, the n! arrays obtained filling in the integers {1, …, n} (without repetitions) in the boxes of [α] are called the αtableaux. To each α-tableau T is associated a primitive idempotent e_{T} of CS_{n} such that CS_{n}e_{T} is an absolutely simple CS_{n}-module which, up to isomorphism, does not depend on T (only on α). We define the Specht module Z_{α} to be a representative for this family of CS_{n}-modules. Then we have :

Theorem 34 (i) The Specht modules {Z_{α}, α a partition of n} provide a basic set of simple CS_{n}-modules. (ii) (Hook Length Formula)

For any partition α of n,

dim Z_{α} = \frac{n!}{∏ (hook lengths)}

where the product is taken over all the boxes of the Young diagram [α].

For a complete description of the Specht modules, we refer to [4]; for the Hook Length Formula, we refer to [5].

3.1.3 Blocks of defect one

The theory of blocks of defect one can be described precisely. For general results, we refer to Goldschmidt ([6]), and we will only give results in the special case of principal blocks of defect one (which is the case we are interested in here).

We let H be a group, B be a p-block of H of defect one, and we let δ(B) =< π >, C_{π} = C_{H}(π) and N_{π} = N_{G}(< π >) (then C_{π} ◁ N_{π}).

The idea is to use the Brauer correspondence, which maps some block b of C_{π} to b^{H} = B, and to lift irreducible characters of b to B. When B is the principal block of H, b is the principal block of C_{π}, and we have :

Theorem 35 If we let e = [N_{π}: C_{π}], then

|IBr_{p}(B)| = e.

We will apply this to H = G and H = N_{G}(P), and their principal blocks B_{0} and b_{0}. In both cases, we get N_{π} = N_{G}(P) and C_{π} = C_{G}(P). By Sylow’s theorems, we have |N_{π}| = p(p − 1) (cf 3.1.5), and, by the Orbit-Stabilizer Theorem, |C_{G}(P)| = \frac{|N_{G}(P)|}{p−1} = p (so that C_{G}(P) = P). Whence e = p − 1 and

|IBr_{p}(B_{0})| = p − 1 = |IBr_{p}(b_{0})|.

We also have a very good description of the decomposition matrix of a block of defect one, which, in our case, reduces to :

Theorem 36 If B = B_{0} or b_{0}, then the entries of the decomposition matrix D_{B} of B are 0’s and 1’s. Moreover, each character φ ∈ IBr_{p}(B) is a constituent of exactly two of the characters in Irr(KB).

Finally, we have an interesting result about real characters :

Theorem 37 If B = B_{0} or b_{0}, and if χ ∈ Irr(KB) is real, then χ has at most two real Brauer constituents.

3.1.4 Principal block of G = S_{p}

Irreducible characters

Take any ordinary irreducible character χ of G. It is associated to a partition α of p and has degree

χ(1) = dim Z_{α} = \frac{p!}{∏ (hook lengths)}.

If p | χ(1), then χ has defect 0 (since |G|_{p} = p), so that {χ} is itself a block of Irr(KG). In particular, elements of the principal block B_{0} of G must have degree prime to p. But, by the Hook Length Formula, (dim Z_{α}, p) = 1 if and only if there exists a hook of length p, which is the maximal possible length for a hook. Thus (dim Z_{α}, p) = 1 if and only if the Young diagram [α] is itself a hook.

This gives us exactly p candidates for belonging to the principal block B_{0}, corresponding to the Young diagrams

⋯

However, since p | |G|, |IBr_{p}(B_{0})| < |Irr(KB_{0})|, so that, via the results on blocks of defect one,

p − 1 < |Irr(KB_{0})| ≤ p

and Irr(KB_{0}) consists exactly of the characters associated to the Young diagrams listed above.

Decomposition matrix

We want to compute the decomposition matrix D_{B_{0}} of B_{0}. First note that there are two linear characters in B_{0}, the trivial one and the signature. If p ≠ 2, then they take distinct values on p-regular classes of G (for example, on 2-cycles). Thus, when restricted to these, they give two distinct irreducible Brauer characters of B_{0}. The two corresponding lines of D_{B_{0}} contain 1 one and p − 2 zeros each. If p = 2, this is still true (even though B_{0} has only one irreducible Brauer character) and

D_{B_{0}} = ( \begin{matrix}1 \\ 1 \end{matrix} ) .

Now all the elements of Irr(KB_{0}) are real-valued, and (as linear combinations with integer coefficients of them) so are the elements of IBr_{p}(B_{0}).

Hence, by Theorem 35, any element of Irr(KB_{0}) has at most two Brauer constituents, and thus there are at most two 1’s in each line of D_{B_{0}}.

However, by Theorem 34, each column of D_{B_{0}} contains two 1’s, so that D_{B_{0}} contains 2(p − 1) ones. Excluding the two lines we mentioned above, there remain 2p − 4 ones to put in p − 2 lines, with at most 2 ones per line. Hence there is only one possibility : each line and each column of D_{B_{0}} contain exactly two 1’s, and, up to reordering lines and columns

D_{B_{0}} = ( \begin{matrix}1 \\ 1 & 1 & & & & & (0) \\ & 1 & 1 \\ & & & . & . & . \\ . & . & . \\ & (0) & & & & & & 1 & 1 \\ & & & & & & & & 1 \end{matrix} ) .

We can label the lines of D_{B_{0}} by the elements χ_{1}, …, χ_{p} of Irr(KB_{0}). The only ones which restrict to elements of IBr_{p}(B_{0}) are thus χ_{1} = 1 (the trivial character) and χ_{p} = ε (the signature).

The Brauer graph of B_{0} is

•  •  •  •  • … 1  χ_{2}  χ_{p−1}  ε

Note that we needed not specify the labelling of χ_{2}, …, χ_{p−1}.

3.1.5 Structure of N = N_{G}(P)

We will first exhibit a cyclic subgroup of N of order p−1. We consider G = S_{p} as the group of permutations of {1, …, p}. Without loss of generality, we can suppose that P =< τ >=< (1 2 … p) >.

For any σ ∈ G, we have

σ^{−1}(1 2 … p)σ = (1^{σ} 2^{σ} … p^{σ}).

We can therefore define σ_{2} ∈ N by letting

σ_{2} : ( \begin{matrix}1 & 2 & … & \frac{p+1}{2} & \frac{p+3}{2} & … & p \\ 1 & 3 & … & p & 2 & … & p − 1 \end{matrix} )

and we have σ_{2}^{−1}τσ_{2} = τ^{2}.

If, for n ∈ N, we denote by [n]_{p} the class of n modulo p ([n]_{p} ∈ {1, …, p}), then

σ_{2} : i ⟼ [2i − 1]_{p} , 1 ≤ i ≤ p.

Similarly, for any k ∈ N, 2 ≤ k ≤ p − 1, we define σ_{k} ∈ G by

σ_{k}: i ⟼ [ki − (k − 1)]_{p} , 1 ≤ i ≤ p.

Then σ_{k}^{−1}τσ_{k} = τ^{k}, so that σ_{k} ∈ N. Note also that σ_{k} fixes 1.

Moreover, for all n ∈ N (n ≠ 0), k ∈ N, 2 ≤ k ≤ p − 1,

σ_{k}^{n}: i ⟼ [k^{n}i − (k^{n} − 1)]_{p} , 1 ≤ i ≤ p.

If n ≥ 1 and 2 ≤ k ≤ p − 1, we get

\begin{aligned}σ_{k}^{n}(2) = 2 & ⟺ & k^{n}2 − k^{n} + 1 ≡ 2 (p) \\ & ⟺ & k^{n} ≡ 1 (p). \end{aligned}

Thus, if we choose k to be a primitive root modulo p, then we have

σ_{k}^{n}(2) = 2 ⟺ ϕ(p) = p − 1 | n

(where ϕ denotes Euler’s indicator function).

Hence (since σ_{k} fixes 1), σ_{k} is a (p − 1)-cycle, and C :=< σ_{k} >≤ N, C cyclic subgroup of order p − 1.

In the sequel, we will write σ for σ_{k}.

Now P ◁ N, so that P C is a subgroup of N, and P ∩ C = {(1)}, so that |P C| = |P | |C| = p(p − 1). However, any Sylow p-subgroup of G = S_{p} is generated by a p-cycle, so that

|Syl_{p}(G)| = \frac{|\{p-cycles\}|}{p − 1} = \frac{p!}{p − 1} = (p − 2)!.

And thus, by Sylow’s theorems,

|N| = |N_{G}(P)| = \frac{|G|}{|Syl_{p}(G)|} = \frac{p!}{(p − 2)!} = p(p − 1).

Hence

N = P C = CP.

3.1.6 Ordinary irreducible characters of N

Conjugacy classes of N

We write π: N = CP ⟶ C the canonical surjective homomorphism. For any c, c′ ∈ C, if there exists g ∈ N such that c = g^{−1}c′g, then

c = π(c) = π(g^{−1}c′g) = π(g^{−1})π(c′)π(g) = π(g)^{−1}c′π(g)

so that c and c′ are also conjugated in C. But C is abelian, so that c = c′. This gives us p − 1 distinct conjugacy classes in N :

Cl_{N}((1)), Cl_{N}(σ), …, Cl_{N}(σ^{p−2}).

The p-cycle τ is clearly not conjugated to any of (1), σ, …, σ^{p−2}, so that Cl_{N}(τ) is another conjugacy class of N.

Now, for any i, 1 ≤ i ≤ p − 2, since C_{N}(σ^{i}) ⊃ C (cyclic of order p − 1), if p | |C_{N}(σ^{i})|, then C_{N}(σ^{i}) = N, which is impossible (C_{N}(σ^{i}) = N would imply σ^{i} ∈ C_{N}(τ); but C_{N}(τ) ⊃ P (cyclic of order p) and |Cl_{N}(τ)| = p − 1 = |P \ {(1)}| so that C_{N}(τ) = P).

Thus, for any i, 1 ≤ i ≤ p − 2, C_{N}(σ^{i}) = C and |Cl_{N}(σ^{i})| = p. This gives us p(p − 2) elements, plus the p elements of P = Cl_{N}((1)) ∪ Cl_{N}(τ). Hence {Cl_{N}((1)), Cl_{N}(σ), …, Cl_{N}(σ^{p−2}), Cl_{N}(τ)} is a complete set of conjugacy classes in N, and |Irr(KN)| = p.

Irreducible characters of N

The group C, cyclic of order p − 1, has p − 1 ordinary irreducible characters, and all are linear. Hence we can lift them to N via the homomorphism π to get p − 1 linear (and thus irreducible) distinct characters of N

ψ_{1} = λ_{1} ∘ π, …, ψ_{p−1} = λ_{p−1} ∘ π,

where Irr(KC) = {λ_{1}, …, λ_{p−1}}.

Since |Irr(KN)| = p, just one character ψ_{p} is missing, and, as

|N| = p(p − 1) = ∑_{ψ∈Irr(KN)} ψ(1)^{2} = (p − 1)1 + ψ_{p}(1)^{2},

ψ_{p} has degree p − 1.

3.1.7 Principal block

Irreducible characters

The ψ_{i}’s, 1 ≤ i ≤ p − 1, are linear, so that for all i, 1 ≤ i ≤ p − 1, ψ̂_{i} ∈ IBr_{p}(N). Furthermore, for all i, 1 ≤ i ≤ p − 1, ψ|_{C} = λ_{i}, so that the ψ_{i}’s separate the conjugacy classes of C; but these classes consist of p-regular elements of N. Thus, if 1 ≤ i, j ≤ p − 1 and i ≠ j, then ψ̂_{i} ≠ ψ̂_{j}.

Writing b_{0} for the prinicipal block of N, δ(b_{0}) = 1 and p | |N| imply

p − 1 = |IBr_{p}(b_{0})| < |Irr(Kb_{0})| ≤ |Irr(KN)| = p.

Hence

IBr_{p}(b_{0}) = {ψ̂_{i}, 1 ≤ i ≤ p − 1}

and

Irr(Kb_{0}) = {ψ_{i}, 1 ≤ i ≤ p}.

Decomposition matrix

It is now easy to compute the decomposition matrix D_{b_{0}} of b_{0}. Indeed, labelling the columns of D_{b_{0}} by ψ̂_{1}, …, ψ̂_{p−1} and its lines by ψ_{1}, …, ψ_{p−1}, ψ_{p}, the first p − 1 lines contain 1 one and p − 2 zeros each. And, since each column of D_{b_{0}} must contain two 1’s, the last line (corresponding to ψ_{p}) must contain only 1’s. Therefore

ψ̂_{p} = ∑_{i=1}^{p−1} ψ̂_{i}

and

D_{b_{0}} = ( \begin{matrix}1 \\ & ... & (0) \\ & (0) & ... \\ & & & 1 \\ 1 & ⋯ & ⋯ & 1 \end{matrix} ) .

3.2 A perfect isometry

3.2.1 Construction

We have

D_{b_{0}} = ψ_{1} ; ⋮ ; ⋮ ; ⋮ ; ψ_{p} ( \begin{matrix}1 \\ & ... & (0) \\ & (0) & ... \\ & & & 1 \\ 1 & ⋯ & ⋯ & 1 \end{matrix} ) and D_{B_{0}} = χ_{1} ; ⋮ ; ⋮ ; ⋮ ; ⋮ ; χ_{p} ( \begin{matrix}1 \\ 1 & 1 & & & & & (0) \\ & 1 & 1 \\ & & & . & . & . \\ . & . & . \\ & (0) & & & & & & 1 & 1 \\ & & & & & & & & 1 \end{matrix} ) .

We define a bijective isometry

I_{p}: ZIrr(Kb_{0}) ⟶ ZIrr(KB_{0})

by letting

I_{p}: ( ψ_{1} ; ψ_{2} ; ψ_{3} ; ⋮ ; ψ_{p−1} ; ψ_{p} ) ⟼ ( χ_{1} ; −χ_{2} ; χ_{3} ; ⋮ ; −χ_{p−1} ; −χ_{p} )

if p is odd. If p = 2, we let similarly

I_{p}: ( \begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix} ) ⟼ ( \begin{matrix}χ_{1} \\ −χ_{2} \end{matrix} ) .

This is not arbitrary. We are led to consider this because we want each column of D_{b_{0}} to be sent to a linear combination of columns of D_{B_{0}} (cf Proposition 27 (iii)).

If we denote by c_{1}, …, c_{p−1} and C_{1}, …, C_{p−1} the columns of D_{b_{0}} and D_{B_{0}} respectively, then we have

I_{p}: { c_{1} ⟼ C_{1} − C_{2} + C_{3} ⋯ − C_{p−1} ; c_{2} ⟼ −C_{2} + C_{3} ⋯ − C_{p−1} ; ⋮ ; c_{p−1} ⟼ −C_{p−1}

Using the same notations as in the proof of Proposition 27, we have

D_{b_{0}} = UD_{B_{0}}V

where

U = ( \begin{matrix}1 \\ & −1 \\ & & 1 & & & & (0) \\ & & & . & . & . \\ & & (0) & & & & ... \\ & & & & & & & −1 \\ & & & & & & & & −1 \end{matrix} ) and V = ( \begin{matrix}1 \\ −1 & −1 & & (0) \\ 1 & 1 & 1 \\ ⋮ & & & ... \\ −1 & −1 & −1 & ⋯ & −1 \end{matrix} ) .

We check that V is invertible in M_{p−1,p−1}(Z) and that

V^{−1} = ( \begin{matrix}1 \\ −1 & −1 & & & (0) \\ & 1 & 1 \\ & & ... & ... \\ & & & ... & ... \\ & (0) & & & −1 & −1 \end{matrix} ) .

Thus the inverse R_{p} of I_{p} maps each column of D_{B_{0}} to a linear combination with integer coefficients of the columns of D_{b_{0}}.

3.2.2 A virtual character

We have seen that Irr(Kb_{0}) = Irr(KN), so that we have a homomorphism

I_{p}: ZIrr(KN) ⟶ ZIrr(KG).

Thus, by Proposition 26, there exists a virtual character μ of G × N such that I_{p} = I_{μ}.

The adjoint R_{μ} of I_{μ} is entirely determined by

< R_{μ}(χ), ζ >_{N}=< χ, I_{μ}(ζ) >_{G} for all χ ∈ Irr(KG), ζ ∈ Irr(KN)

so that we see that

R_{μ} |_{ZIrr(KB_{0})} = R_{p}.

We intend to show that I_{μ} defines a perfect isometry between b_{0} and B_{0}. Rather than trying to compute μ and check that it is perfect, we will use Proposition 28, whose notations are now in force.

3.2.3 Proposition 28 (ii)

The (Φ_{ψ})_{ψ∈IBr_{p}(b_{0})} form a basis of CF_{p′}(N, b_{0}, K). Similarly, the (Φ_{φ})_{φ∈IBr_{p}(B_{0})} form a basis of CF_{p′}(G, B_{0}, K). They are represented by the columns of D_{b_{0}} and D_{B_{0}} respectively. Hence we have already seen that, by construction,

I_{μ}: CF_{p′}(N, b_{0}, K) ⟶ CF_{p′}(G, B_{0}, K)

and

R_{μ}: CF_{p′}(G, B_{0}, K) ⟶ CF_{p′}(N, b_{0}, K).

3.2.4 Conjugacy classes of N and G

For C a conjugacy class of N (resp. G), we define α_{C} (resp. β_{C}) on Kb_{0} = KNε_{0} (resp. KB_{0} = KGe_{0}) by

α_{C}(n) = 1_{Cε_{0}}(nε_{0}) for all n ∈ N

and

β_{C}(g) = 1_{Ce_{0}}(ge_{0}) for all g ∈ G.

The sets {α_{C}, C conjugacy class of N} and {β_{C}, C conjugacy class of G} are O-bases for CF (N, b_{0}, O) and CF (G, B_{0}, O) respectively (the ones we used in the proof of Proposition 28).

The α_{C}’s can be written in the K-basis of CF (N, b_{0}, K) consisting of the elements of Irr(Kb_{0}) : for any class C of N,

α_{C} = ∑_{ψ∈Irr(Kb_{0})} < ψ, α_{C} >_{N} ψ.

And, for all ψ ∈ Irr(Kb_{0}),

< ψ, α_{C} >_{N} = \frac{1}{|N|} ∑_{g∈N} ψ(g)\overline{α_{C}(g)}
= \frac{1}{|N|} ∑_{g∈C} ψ(g)
= \frac{|C|}{|N|}ψ(C).

In particular, whenever C is such that |N|_{p} = p | |C|, < ψ, α_{C} >_{N}∈ O for all ψ ∈ Irr(Kb_{0}) (since ψ(C) ∈ O), so that

I_{μ}(α_{C}) ∈ CF (G, B_{0}, O).

The classes Cl_{N}(σ), …, Cl_{N}(σ^{p−1}) all have order p, thus

I_{μ}(α_{C}) ∈ CF (G, B_{0}, O) for all C ∈ {Cl_{N}(σ), …, Cl_{N}(σ^{p−1})}.

Remain to study α_{{(1)}} =: α_{1} and α_{Cl_{N} (τ)} =: α_{τ}.

Similarly, whenever C is a conjugacy class of G such that |G|_{p} = p | |C|, < χ, β_{C} >_{G}∈ O for all χ ∈ Irr(KB_{0}), so that

R_{μ}(β_{C}) ∈ CF (N, b_{0}, O).

Suppose that C is a class of G such that p |̸ |C|, and g ∈ C. Then p | |C_{G}(g)|, so that g ∈ C_{G}(ρ) for some p-cycle ρ ∈ C_{G}(g). But C_{G}(ρ) =< ρ >, so that g is either (1) or a p-cycle.

Thus, if C is neither {(1)} nor consisting of (the) p-cycles, then

R_{μ}(β_{C}) ∈ CF (N, b_{0}, O).

It remains to study β_{{(1)}} =: β_{1} and β_{Cl_{G}(τ)} =: β_{τ}.

3.2.5 Values on p-elements

In order to establish that R_{μ} and I_{μ} satisfy Proposition 28 (i), we prove two important results about them. Namely

Theorem 38 Writing I_{p} = I_{μ} and R_{p} = R_{μ}|_{ZIrr(KB_{0})},

(1) I_{p} and R_{p} preserve the degree (value on 1) of irreducible characters (of b_{0} and B_{0} respectively) modulo p.

(2) I_{p} and R_{p} preserve the value of irreducible characters on p-cycles (of N and G respectively).

Proof : The proofs of (1) and (2) are entirely based on Proposition 6 (ii) and (i) respectively, and on the structure of the Brauer Graph of B_{0}.

We describe the behavior of Irr(Kb_{0}) and Irr(KB_{0}) on 1 and the p-cycles.

The characters ψ_{1}, …, ψ_{p−1} of N come from linear characters of C; thus they are linear and have P in their kernel :

ψ_{i}(1) = 1, ψ_{i}(τ) = 1 for all i, 1 ≤ i ≤ p − 1.

Now ψ̂_{p} = ψ̂_{1} + ⋯ + ψ̂_{p−1}, so that

ψ_{p}(1) = p − 1 ≡ −1 (p).

Moreover, by Proposition 6 (i), Φ_{ξ_{1}}(τ) = 0 (where ξ_{1} = ψ̂_{1} ∈ IBr_{p}(b_{0})). D_{b_{0}} shows that Φ_{ξ_{1}} = ψ_{1} + ψ_{p}, so that

ψ_{p}(τ) = −ψ_{1}(τ) = −1.

If φ_{1} = 1, φ_{2}, …, φ_{p−1} ∈ IBr_{p}(B_{0}) correspond to the columns of D_{B_{0}}, we have Φ_{φ_{1}} = 1 + χ_{2}, Φ_{φ_{2}} = χ_{2} + χ_{3}, …, Φ_{φ_{p−1}} = χ_{p−1} + ε. And, by Proposition 6 (i) and (ii), for all φ ∈ IBr_{p}(B_{0}),

Φ_{φ}(τ) = 0 and |G|_{p} = p | Φ_{φ}(1).

Therefore, starting with the trivial character and its values on 1 and τ, we can follow the Brauer Graph of B_{0} to compute the values of the other irreducible characters of B_{0} :

χ_{1}(1) = 1 and χ_{1}(τ) = 1.

Then Φ_{φ_{1}} ≡ 0 (p) and Φ_{φ_{1}}(τ) = 0 respectively imply

χ_{2}(1) ≡ −1 (p) and χ_{2}(τ) = −1.

Using similarly Φ_{φ_{2}}, …, Φ_{φ_{p−1}}, we get

χ_{i}(1) ≡ (−1)^{i+1} (p) and χ_{i}(τ) = (−1)^{i+1} for all i, 1 ≤ i ≤ p.

Now, since

I_{p}: \{ \begin{matrix}ψ_{i} ⟼ (−1)^{i+1}χ_{i} for all i, 1 ≤ i ≤ p − 1 \\ ψ_{p} ⟼ −χ_{p} \end{matrix}

we see that, for all i, 1 ≤ i ≤ p,

I_{p}(ψ_{i})(1) ≡ ψ_{i}(1) (p) and I_{p}(ψ_{i})(τ) = ψ_{i}(τ).

Similarly, R_{p} preserves the degree modulo p and the value on p-cycles of the elements of Irr(KB_{0}). This ends the proof.

Remark : We have pointed out that (I1) I_{p} is an isometry (I2) I_{p} preserves the character degrees modulo p (I3) I_{p} preserves the values on the p-singular p-elements.

This expresses the fact that I_{p} is an isotypy, as presented in [3], between B_{0} and b_{0}.

3.2.6 Back to 3.2.4

We have

I_{μ}(α_{1}) = \frac{1}{|N|} ∑_{ψ∈Irr(Kb_{0})} ψ(1)I_{μ}(ψ)

and

R_{μ}(β_{1}) = \frac{1}{|G|} ∑_{χ∈Irr(KB_{0})} χ(1)R_{μ}(χ).

But, for any ψ ∈ Irr(Kb_{0}), I_{μ}(ψ) = ±χ for some χ ∈ Irr(KB_{0}), and, by Theorem 38 (1)

ψ(1) ≡ I_{μ}(ψ)(1) ≡ ±χ(1) (p)

(and χ takes integer values).

Thus, for all g ∈ G, ∑_{χ∈Irr(KB_{0})} χ(1)χ(g) ∈ Z, and (since I_{p} is a bijection)

∑_{ψ∈Irr(Kb_{0})} ψ(1)I_{μ}(ψ)(g) ≡ ∑_{χ∈Irr(KB_{0})}(±χ(1))(±χ(g)) (p)
≡ ∑_{χ∈Irr(KB_{0})} χ(1)χ(g) (p)
≡ ∑_{χ∈Irr(KG)} χ(1)χ(g) (p)

(since all irreducible characters of G outside B_{0} have degree divisible by p). Hence

∑_{ψ∈Irr(Kb_{0})} ψ(1)I_{μ}(ψ)(g) ≡ |G|β_{1}(g) ≡ 0 (p).

PART 3. BROUÉ’S CONJECTURE 45 So that (since |N|_{p} = p) I_{μ}(α_{1})(g) ∈ O for all g ∈ G, and

I_{μ}(α_{1}) ∈ CF (G, B_{0}, O).

Similarly

∑_{χ∈Irr(KB_{0})} χ(1)R_{μ}(χ)(g) ≡ ∑_{ψ∈Irr(Kb_{0})} ψ(1)ψ(g) = ∑_{ψ∈Irr(KN)} ψ(1)ψ(g) ≡ 0 (p)

for all g ∈ G. Whence (since |G|_{p} = p)

R_{μ}(β_{1}) ∈ CF (N, b_{0}, O).

Now

\frac{|Cl_{N}(τ)|}{|N|} = \frac{1}{|C_{N}(τ)|} = \frac{1}{|C_{G}(τ)|} = \frac{|Cl_{G}(τ)|}{|G|} = \frac{1}{p}

so that

I_{μ}(α_{τ}) = \frac{1}{p} ∑_{ψ∈Irr(Kb_{0})} ψ(τ)I_{μ}(ψ)

and

R_{μ}(β_{τ}) = \frac{1}{p} ∑_{χ∈Irr(KB_{0})} χ(τ)R_{μ}(χ).

Then, as above, by Theorem 38 (2), for all g ∈ G,

∑_{ψ∈Irr(Kb_{0})} ψ(τ)I_{μ}(ψ)(g) = ∑_{χ∈Irr(KB_{0})}(±χ(τ))(±χ(g))
= ∑_{χ∈Irr(KB_{0})} χ(τ)χ(g)
= ∑_{χ∈Irr(KG)} χ(τ)χ(g)

(since, by Proposition 11, every element of Irr(KG)\Irr(KB_{0}), having defect 0, vanishes on τ (which is p-singular)). Thus

∑_{ψ∈Irr(Kb_{0})} ψ(τ)I_{μ}(ψ)(g) = \{ \begin{matrix}|C_{G}(τ)| & if g is a p-cycle \\ 0 & otherwise \end{matrix}
≡ 0 (p).

Hence, for all g ∈ G, I_{μ}(α_{τ})(g) ∈ O, and

I_{μ}(α_{τ}) ∈ CF (G, B_{0}, O).

Similarly, for all g ∈ G,

∑_{χ∈Irr(KB_{0})} χ(τ)R_{μ}(χ)(g) = ∑_{ψ∈Irr(Kb_{0})} ψ(τ)ψ(g) = ∑_{ψ∈Irr(KN)} ψ(τ)ψ(g) ≡ 0 (p).

Hence, for all g ∈ G, R_{μ}(β_{τ})(g) ∈ O, and

R_{μ}(β_{τ}) ∈ CF (N, b_{0}, O).

3.2.7 The solution

Putting together the results of 3.2.4 and 3.2.6, we have

(i) I_{μ} maps CF (N, b_{0}, O) into CF (G, B_{0}, O) and R_{μ} maps CF (G, B_{0}, O) into CF (N, b_{0}, O); and, by 3.2.3, (ii) I_{μ} maps CF_{p′}(N, b_{0}, K) into CF_{p′}(G, B_{0}, K) and R_{μ} maps CF_{p′}(G, B_{0}, K) into CF_{p′}(N, b_{0}, K).

Therefore, by Proposition 28, the character μ is perfect. Since, by construction, I_{μ} induces a bijective isometry between ZIrr(Kb_{0}) and ZIrr(KB_{0}), I_{μ} is a perfect isometry between B_{0} and b_{0}.

We have thus proved

Theorem 39 If p is a prime and N is the normalizer of a Sylow p-subgroup of S_{p}, then the principal blocks of S_{p} and N are perfectly isometric.

Conclusion

After having introduced the ideas of Block Theory, the study of defect groups and their behaviour under the Brauer correspondence revealed to us a first link between the principal blocks of a group and of the normalizer of one of its Sylow p-subgroups. Via the concept of perfect isometries, we deepened this link to the level of characters and idempotents of the group algebra.

A perfect isometry between two blocks is a correspondence between characters, induced by an equivalence between the corresponding derived categories. An isotypy between blocks must reveal an even deeper equivalence between derived categories (see [1]).

In this sense, the proof we gave in a very special case of Brou’s Conjecture is a first step in the understanding of the structure and behaviour of derived categories of blocks.

I hope to be able to understand the other steps through my thesis work...

Acknowledgements

The idea of this mmoire comes from Pr. Geoffrey R. Robinson (The University of Birmingham), and most of its content was elaborated with the precious help of Meinolf Geck.

I wish to thank M. Geck for the time he gave me, to supervise and advise my work, as well as to answer my (numerous) questions all along the year. The first part of this mmoire is mainly based on the lectures he gave on modular representation theory during the first semester of this year.

Many thanks also to G. Robinson, who taught me the ordinary representation theory during my masters, and accepted to direct my (future) work.

References

[1] M. Brou, Isomtries parfaites, types de blocs, catgories drives, Astrisque 181-182 (1990), pp. 61-92.

[2] M. Brou, Equivalences of blocks of group algebras, Proceedings of the International Conference on Representations of Algebras, Ottawa (1992).

[3] M. Brou, Rickard Equivalences and Block Theory, Proceedings of the International Conference on Group Theory ”Groups 93”, Galway (1993).

[4] C. Curtis, I. Reiner, ”Methods of representation theory”, vol. 2, Wiley (1987).

[5] W. Fulton, J. Harris, ”Representation Theory - a first course”, Graduate Texts in Mathematics 129, Springer.

[6] David M. Goldschmidt, ”Lectures on Character Theory”, Mathematics Lectures series 8, Publish or Perish, Inc. (1980).