Generalized Perfect Isometries in some Groups of Lie Rank One

Jean-Baptiste Gramain

Institut Girard Desargues Université Claude Bernard, Lyon 1

September 2004

Abstract In a recent paper, B. Külshammer, J. B. Olsson and G. R. Robinson introduced notions of generalized blocks and generalized perfect isometries, and studied them in the case of the symmetric group S_{n}. It is the purpose of this paper to investigate some families of groups of Lie rank one, exhibiting generalized perfect isometries where there are none in the sense given by M. Broué, and thus proving a (weaker) analogue of one of Broué’s Conjectures in these cases.

1 Introduction

For the definitions we give in this introduction, we refer to [9].

1.1 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. Now let us fix a closed set C of conjugacy classes of G, and assume that 1 ∈ 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, χ and ψ are said to be orthogonal across C. We call < χ, ψ >_{C} the C-contribution associated to χ and ψ. Direct C-linking is a symmetric and reflexive (since 1 ∈ C) binary relation on Irr(G). Extending it by transitivity to an equivalence relation, we obtain a partition of Irr(G) into C-blocks.

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. In [9], B. Külshammer, J. B. Olsson and G. R. Robinson have defined ℓ-blocks for the symmetric groups, for ℓ an arbitrary integer, and proved an ℓ- analogue of the Nakayama Conjecture. In this article, we will take C to be the set of p^{k}-regular elements of G, where k ≥ 1 and g ∈ G is said to be p^{k}-regular if its order is not divisible by p^{k}, and p^{k}- singular otherwise. Note that, then, C is a union of p-sections of G, so that, by Brauer’s Second main Theorem, two characters belonging to distinct p-blocks of G are orthogonal across C (since they are orthogonal across each p-section). Thus any p-block of G is a union of p^{k}-blocks.

1.2 Perfect isometries

We find in [9] 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 (see [1]).

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 1.1. (see [1]) A generalized character μ of G × H, is perfect if the following two conditions are satisfied: (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. 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 (see [1]):

Theorem 1.1. 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 modular irreducible 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 [9]. 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 ∈ Irr(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 that R(C, b) and P(C, b) have the same Z-rank (see [9]). 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 Zbasis 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 (see [9]):

Theorem 1.2. 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 perfect isometry induces a bijection between the pblocks in b and b′, which preserves the numbers of ordinary irreducible characters and modular irreducible characters, and the elementary divisors of the Cartan matrix. However, with this more general definition, one doesn’t get the algebra isomorphism of Theorem 1.1 (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 groups of Lie rank one when p is the defining characteristic, for then the Sylow p-subgroups are not Abelian. Furthermore, it is shown in [4] 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 article, 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.

The first section deals with special unitary groups SU(3, q^{2}), the second with Suzuki groups Sz(q), and the third with Ree groups (of type G_{2}) Re(q).

1.3 Some notations and results

In all the sequel, we will use the following notations: if G is a finite group, H ≤ G a subgroup of G, and x ∈ G, then C_{G}(x) (resp. C_{G}(H)) is the centalizer of x (resp. H) in G, Cl_{G}(x) is the conjugacy class of x in G, Cl_{G}(H) is the set of the conjugacy classes in G of the elements of H, N_{G}(H) is the normalizer of H in G. If p is a prime, O_{p}(G) is the largest normal p-subgroup of G.

If χ and ψ are complex class functions of G and H respectively, then Ind_{H}^{G}(ψ) and Res_{H}^{G}(χ) are the class function induced from H to G by ψ and the restriction from G to H of χ respectively. We will also write Res_{H}^{G}(χ) = χ|_{H}. For χ and φ complex class functions of G, < χ, φ >_{G} denotes the scalar product: < χ, φ >_{G}= \frac{1}{|G|} ∑_{x∈G} χ(x)\overline{φ(x)}. For x ∈ G, we will sometimes write ∘(x) for the order of x.

In the three 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. [8], Corollary (4.21)). An immediate consequence of this is that, if the Sylow p-subgroups of G have trivial intersection, then any p-block 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. [7], 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), then G has exactly one p-block.

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

In this section, we denote by G = SU(3, q^{2}) the 3-dimensional special unitary group on F_{q^{2}}, where q = 2^{n}, n ≥ 1. We write d = gcd(3, q +1). Then the center Z(G) of G is cyclic of order d. The order of G is |G| = q^{3}(q+1)^{2}(q−1)(q^{2}−q+1). We consider U a Sylow 2-subgroup of G and its normalizer B = N_{G}(U), which is the semidirect 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 [6], whose notations are now in force.

2.1 Conjugacy classes

Using the parametrization of conjugacy classes of G and B given in [6], we obtain that: G has q^{2} + q + 1 + d^{2} conjugacy classes, d^{2} of which are classes of elements of order divisible by 4. These are the C_{3}^{(k,l)}, 0 ≤ k, l ≤ d − 1, which have length \frac{1}{d}q(q^{2} − 1)(q^{3} + 1) and whose elements have order 4 if k = 0 and 4d otherwise. In addition, G has 1 conjugacy class of involutions of length (q^{3} + 1)(q − 1). B has q^{2} + q + d conjugacy classes, d^{2} of which are classes of elements of order divisible by 4. These are the B_{3}^{(k,l)}, 0 ≤ k, l ≤ d−1, which have length \frac{1}{d}q(q^{2}−1) and which fuse in the C_{3}^{(k,l)}’s into G.

2.2 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 [6]); 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 2-blocks B_{0} and b_{0} of G and B respectively. Writing G_{2} the set of all 2-elements of G, we have |G_{2}| = 1 + q(q^{2} − 1)(q^{3} + 1) + (q^{3} + 1)(q − 1) = (q^{3} + 1)(q^{3} − 1) + 1. On the other hand, writing Syl_{2}(G) the set of Sylow 2-subgroups of G, we have

|Syl_{2}(G)|(|U| − 1) + 1 = [G : B](|U| − 1) + 1 = (q^{3} + 1)(q^{3} − 1) + 1.

This proves that any two Sylow 2-subgroups of G must either be equal or have trivial intersection, i.e. ∀g ∈ G \ B , U ∩ gUg^{−1} = 1. This implies (as indicated in the introduction) that any 2-block of G must have either full defect or defect 0.

If d = 1, then, for any x ∈ B_{3}^{(0,l)} ⊂ U, |B_{3}^{(0,l)}| = q(q^{2} − 1), so that |C_{B}(x)| = q^{2} (a power of 2), and thus C_{B}(x) ≤ U. Hence C_{B}(U) ≤ U and, since U is the largest normal 2-subgroup of B, this implies that B has only one 2-block. Brauer’s First Main Theorem now implies that any non-principal 2-block of G has defect 0. Since G has only one character of 2-defect 0, χ_{q^{3}}, we obtain that,

if d = 1, then b_{0} = Irr(B) and B_{0} = Irr(G) \ {χ_{q^{3}}}.

In particular, if d = 1, then |B_{0}| = |b_{0}| = q^{2} + q + 1.

If d = 3, then Z(G) ⊂ B = N_{G}(U), so that Z(G) ⊂ C_{B}(U) and this argument cannot be applied anymore. We therefore compute directly B_{0} and b_{0}, using the fact that two (complex) irreducible characters belong to the same pblock if and only if the associated central characters are the same when reduced to a field of characteristic p (cf [8], definition (3.1) and Problem (3.3)). For χ ∈ Irr(G), the associated central character ω_{χ}: Z(CG) ⟶ C is defined by: if C is a conjugacy class of G, x_{C} ∈ C and Ĉ = ∑_{x∈C} x, then

ω_{χ}(Ĉ) = \frac{|C|χ(x_{C})}{χ(1)}.

The principal blocks B_{0} and b_{0} can thus be computed directly from the character tables of G and B found in [6]. We finally obtain:

if d = 3, then b_{0} = {θ_{1}^{(u)}; 3|u} ∪ {θ_{q(q−1)}^{(u)}; 3|u} ∪ {θ_{(q^{2}−1)/3}^{(0,v)}}
and B_{0} = {χ_{1}, χ_{q^{2}−q}} ∪ {χ_{q^{2}−q+1}^{(u)}, χ_{q(q^{2}−q+1)}^{(u)}; 3|u} ∪ {χ_{q^{3}+1}^{(u)}; 3|u}
∪{χ_{(q−1)(q^{2}−q+1)}^{(u,v)}; 3|u + v} ∪ {χ_{(q+1)^{2}(q−1)}^{(u)}; 3|u} ∪ {χ_{(q−1)(q^{2}−q+1)/3}^{(u)}}

and it can be verified, using the fact that q ≡ −1(mod 3), that, if d = 3, then |b_{0}| = |B_{0}| = q \frac{q+1}{3} + 3.

2.3 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 relabeling 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 re-labeling 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.

3 The case of 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).

3.1 Conjugacy classes

The conjugacy classes of G can be found in [2]. 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 [4].

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, C ≅ U/C ≅ (GF (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. |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.

3.2 Irreducible characters of B

3.2.1 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 nontrivial irreducible characters of U (cf [5] 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 application

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

3.2.2 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}}) ∀c ∈ C, ∀h ∈ B, for 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 [5] 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.

3.2.3 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}

3.2.4 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_{S}(f_{j})|(χ_{k}(f_{j}) + χ_{k}(f_{j}^{−1})) + ∑_{j∈I_{1}} |Cl_{S}(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}(s) = ∑_{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}); i ∈ {1, … 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.

3.2.5 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, noting 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}.

3.3 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 [2], we see that G has exactly one character of defect 0, written Π, and thus B_{0} =Irr(G) \ {Π}.

We take from [2] 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 value 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}

4 The case of 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).

4.1 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 [3]. In particular, we find in [3] 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, |H| = q − 1, H can be written 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 [10]. There can be found the following:

Theorem 4.1. 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}. B = N_{G}(U) = UH and 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 − 3m and q + 1 + 3m (the orders of their centralizers, cf [10], 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 [10], pp 78-80), Y, Y T and Y T^{−1} of order 9 (cf [10], p 82), J of order 2, JT and JT^{−1} which belong to groups whose order has 3-valuation 1 (cf [10], 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.

4.2 Conjugacy classes of U

4.2.1 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}, 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.

4.2.2 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.

4.3 Irreducible characters of U

4.3.1 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 [5] 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).

4.3.2 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 application

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).

4.3.3 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 non-linear 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 appear as irreducible summands of characters induced from W, with multiplicity their 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 relabeling 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}.

4.3.4 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 iff 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 iff 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

???????

4.3.5 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

< 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 Horbits. 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}; i = 1 … 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}; i = 1 … 3q − 3} is a union of an even number of Horbits, 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, relabeling 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, 1 ≤ j ≤ (q − 1)/2} , {χ_{3,j}; 1 ≤ j ≤ (q − 1)/2}
\{\overline{χ_{i,j}}; i = 1, 2, 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.

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

4.4.1 Clifford’s Theory

The following description of Clifford’s Theory can be found in [7]. 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].

4.4.2 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, 1 ≤ j ≤ \frac{q−1}{2}\} ⟶ χ, degree (q − 1)3^{m}
\{\overline{χ_{i,j}}; i = 1, 2, 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.

4.4.3 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, 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.

4.5 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 [10], 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 have the following fragments of character table for B_{0} (cf [10]) 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 relabeling 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.

Acknowledgements: I wish to thank deeply my supervisors Meinolf Geck (Université Claude Bernard, Lyon) and Geoffrey R. Robinson (The University of Birmingham, Birmingham), who gave me these questions and examples to work on, and for all the help and time they gave me in the last two years (and before). Thank you also to Attila Maróti (University of Southern California, Los Angeles) who helped me putting the pieces together.

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