PACIFIC JOURNAL OF MATHEMATICS Vol. 187, No. 2, 1999

FINITE GROUPS OF LIE TYPE OF SMALL RANK

Frauke M. Bleher

In this paper, it is proven that a conjecture of Zassenhaus is valid for all finite simple groups of Lie type of rank 1 and of rank 2 which are not of type ^{2}A_{3} or ^{2}A_{4}. In particular, this conjecture holds for all finite simple groups with abelian Sylow 2–subgroups.

1. Introduction.

One of the key problems in the representation theory of finite groups is to represent a given finite group G in a suitable way. To do so, invariants which classify G up to isomorphisms are determined. One of the questions derived in this context is the so-called Isomorphism Problem (IP), which asks whether the existence of an isomorphism between the integral group algebras ℤG and ℤH implies that G and H are isomorphic. In the seventies, Zassenhaus conjectured a stronger version of (IP) describing the structure of the group of units V (ℤG) of ℤG with augmentation 1:

(ZC). Any two subgroups X, Y of V (ℤG) which have the same order as G are conjugate by a unit of ℚG.

The conjecture (ZC) is not true in general, as was shown by Roggenkamp and Scott who constructed a metabelian counter-example [38, IX §1]. Note that since this counter-example is a metabelian group, (IP) is still valid for this group. Recently, Hertweck also found a counter-example to the general validity of (IP) which is a soluble group [18].

Nevertheless, it still remains an interesting question for which classes of finite groups (ZC) is valid. Roggenkamp and Scott showed that (ZC) is valid for groups whose generalized Fitting subgroup is a p-group [37]. Moreover, Weiss was able to show that for nilpotent groups each finite subgroup of V (ℤG) is conjugate to a subgroup of G by a unit of ℚG [45, 46]. In case G is soluble, Čech style cohomology sets can be used to get obstructions for (ZC) to be true [30].

For finite simple groups however, new methods have to be developed to examine (ZC). In [4] it is proven that (ZC) is valid for all minimal simple groups and for all simple Zassenhaus groups. But it is still an open question whether the conjecture (ZC) is valid for all finite simple groups.

If G is finite simple, then (IP) has always a positive solution [29, Thm. 2.3]. Furthermore, from [5, Thm. 1] it follows that (IP) is valid for all finite groups of Lie type 𝔾^{F}, where 𝔾 is a simply connected simple algebraic group over an algebraically closed field of finite characteristic and F is a Frobenius map in the sense of [8, §1.17]. Note that these groups 𝔾^{F} are certain central extensions of the finite simple groups of Lie type. In general, if (IP) holds for G, (ZC) is equivalent to the following description of the group Aut_{n}(ℤG) of augmentation preserving ring automorphisms of ℤG:

(ZCAut). Every σ ∈ Aut_{n}(ℤG) can be written as σ = τ ∘ α, where α is the ℤ–linear extension of a group automorphism of G and τ ∈ Aut_{n}(ℤG) fixes the class sums of G.

Thus, if (IP) is valid for G, the structure of certain ring automorphisms of ℤG can be used to study the group of units V (ℤG). Since (IP) has a positive solution for all groups considered in this paper, we will always examine (ZCAut) to verify (ZC).

The main results of this paper are the following two theorems:

Theorem 1. The conjecture (ZC) is valid for all finite simple groups with abelian Sylow 2-subgroups.

Theorem 2. The conjecture (ZC) is valid for all finite simple groups of Lie type of rank 1 and of rank 2, which are not isomorphic to the unitary groups PSU(4, q^{2}) and PSU(5, q^{2}).

Note that because of the close relationship of the ordinary and modular representation theory of linear and unitary groups (see [26] and [43]), PSU(4, q^{2}) and PSU(5, q^{2}) should be examined together with the groups PSL(4, q) and PSL(5, q).

To prove these statements we use ordinary and modular representation theory. In particular, modular representations in the defining characteristic as described by Steinberg’s tensor product theorem [43] play an important role. Here the knowledge of the action of σ ∈ Aut_{n}(ℤG) on tensor products as given in [4, Prop. 2.1] is essential. Furthermore, we need to examine the generic ordinary character tables of the considered groups. Except for the groups ^{2}F_{4}(q^{2}), these character tables can be found in [9], [13], [14], [15], [16], [34] ,[41], [42], or [44]. All character tables are also given in the computer algebra program CHEVIE [17] which provides generic ordinary character tables for several series of finite groups of Lie type of small rank. Furthermore, in CHEVIE known character tables have been verified, mistakes have been corrected and the data has been completed.

The article is organized as follows: In Section 2, we state basic results and techniques which are needed to prove the theorems. In particular, we show that if G is a finite simple group of Lie type with defining characteristic p, or a certain central extension, then the action of σ ∈ Aut_{n}(ℤG) on the p-modular character table commutes with the operation induced by field automorphisms. In Section 3, the validity of (ZC) for the finite Ree groups of type F_{4} and G_{2} is proved. This establishes, together with the results of [4], Theorem 1. Section 4 is then dedicated to the proof of Theorem 2. Note that in [4] it is already shown that (ZC) is valid for the groups SL(2, q), PSL(2, q) and ^{2}B_{2}(q^{2}) for all possible prime powers q.

The notations used are mainly standard, see for example [11]. The set of the conjugacy classes of G is denoted by Cl(G), Cl(g) is the conjugacy class of g ∈ G, and Cl(G_{p′}) denotes the set of the conjugacy classes of p-regular elements of G. Suppose (K, R, k) is a p-modular system with K sufficiently large relative to G and char(K) = 0. Thus both K and k are splitting fields for G. Then the set of the ordinary irreducible characters Irr(G) is identified with the set of the characters afforded by simple KG-modules. The irreducible Brauer characters IBr(G) are the irreducible Brauer characters with respect to (K, R, k).

2. Preliminaries.

In this section, we want to provide basic results and techniques needed to prove the Theorems 1 and 2.

Suppose G is an arbitrary finite group and (K, R, k) is a p-modular system with K sufficiently large relative to G. Given σ ∈ Aut_{n}(ℤG), σ induces augmentation preserving algebra automorphisms σ ∈ Aut_{n}(KG), respectively σ ∈ Aut_{n}(kG), because there are ring homomorphisms from ℤ to K, respectively k. Then σ defines an autoequivalence of the category of finitely generated KG-modules, respectively kG-modules, by mapping a module M to M^{σ}. The twisted module M^{σ} is defined to be equal to M as vector space, but g ∈ G operates on M^{σ} as σ(g). Thus σ induces an operation on the ordinary characters, respectively Brauer characters, of G. Because σ also induces a class sum correspondence [38, IV], i.e. σ maps class sums to class sums, σ defines also an action on the conjugacy classes of G. These two actions are compatible in the sense that σ induces a character table automorphism of the ordinary, respectively p-modular, character table of G. This means that for all χ ∈ Irr(G), φ ∈ IBr(G), C ∈ Cl(G) and C′ ∈ Cl(G_{p′})

χ^{σ}(C) = χ(C^{σ})
φ^{σ}(C′) = φ(C′^{σ}).

Furthermore, the operation of σ on characters commutes with tensor products:

Proposition 2.1 ([4, Prop. 2.1]). Let ξ and ζ be two ordinary characters, respectively Brauer characters, then

(ξ ⊗ ζ)^{σ} = ξ^{σ} ⊗ ζ^{σ}.

By [28, Thm. V.1(c)], it follows that σ preserves also the power map in the sense that

(C^{n})^{σ} = (C^{σ})^{n}

for all positive integers n. Note that if C = Cl(g), then C^{n} = Cl(g^{n}).

The operation on the ordinary character table induced by σ ∈ Aut_{n}(ℤG) yields a statement which is equivalent to (ZCAut):

Lemma 2.2. The conjecture (ZCAut) is valid for G if and only if for every σ ∈ Aut_{n}(ℤG) there exists α ∈ Aut(G) such that σ and α induce the same character table automorphism on the ordinary character table of G.

Thus Lemma 2.2 provides a necessary and sufficient criterion for the validity of (ZCAut) which uses only the ordinary character table and its automorphisms.

For the remainder of the paper, we want to concentrate on finite groups which are associated to Chevalley groups of universal type over an algebraically closed field.

Let G_{ℂ} be a simple Lie algebra over ℂ with Cartan-subalgebra H_{ℂ}. Let Φ be the root system of G_{ℂ} with respect to H_{ℂ}, and let Δ = {s_{1}, …, s_{l}} be the corresponding system of fundamental roots.

Suppose k̄ is an algebraically closed field of characteristic p > 0. The simply connected simple algebraic group, i.e. the Chevalley group of universal type, 𝔾 of type Φ over k̄ is generated by {x_{r}(t) | r ∈ Φ, t ∈ k̄} with relations as in [7, Thm. 12.1.1]. If h_{s}(t) = x_{s}(t)x_{−s}(−t^{−1})x_{s}(t)x_{s}(−1)x_{−s}(1)x_{s}(−1), then 𝕋 = ⟨h_{s}(t) | s ∈ Δ, t ∈ k̄⟩ is a maximal torus of 𝔾.

Since 𝔾 is simply connected, the group X = X(𝕋) of rational characters of 𝕋 is the full lattice of weights, with a basis consisting of the fundamental weights {λ_{1}, ⋯, λ_{l}}. Denote by X^{+} the set ∑ ℤ_{0}^{+}λ_{i} of all dominant weights. There exists a natural partial ordering of X given by μ ≤ λ if λ − μ is a sum of positive roots. The group X/pX of restricted weights with respect to p is identified with X_{p} = {∑ c_{i}λ_{i} | 0 ≤ c_{i} ≤ p − 1}. Given λ ∈ X^{+}, λ can be written uniquely as λ = μ_{0} + pμ_{1} + ⋯ + p^{m−1}μ_{m−1} for a suitable m ≥ 1 with μ_{i} ∈ X_{p}. If X_{q} = {∑ c_{i}λ_{i} | 0 ≤ c_{i} ≤ q − 1}, q = p^{m}, then it follows that λ ∈ X_{q}.

Since G_{ℂ} has a Chevalley basis {e_{r}, h_{s} | r ∈ Φ, s ∈ Δ} as ℂ–basis whose structure constants all lie in ℤ, we can define G_{ℤ} to be the ℤ–lattice spanned by the Chevalley basis, and Gk̄ = k̄ ⊗_{ℤ} G_{ℤ}. Then Gk̄ is a Lie algebra over k̄, and we denote 1 ⊗ e_{r} and 1 ⊗ h_{s} again by e_{r} and h_{s}. A Gk̄-module M is called restricted, if e_{r}^{p} · m = 0 and h_{s}^{p} · m = h_{s} · m for all r ∈ Φ, s ∈ Δ and m ∈ M.

The non-isomorphic irreducible restricted Gk̄-modules, which were described by Curtis (see [43, (2.7)]), correspond bijectively to the restricted weights in X_{p}. Let M be a full set of representatives of these modules, and let M_{λ} denote the irreducible restricted Gk̄-module in M which corresponds to λ ∈ X_{p}.

According to [43, §4] the Gk̄-module M_{λ} can be lifted to a simple k̄𝔾– module M_{λ} which is generated by a maximal vector v^{+} of highest weight λ. This means that if we look at the weight spaces M_{λ}(μ) = {m ∈ M_{λ} | x·m = μ(x) · m for all x ∈ 𝕋}, μ ∈ X, then v^{+} ∈ M_{λ}(λ) and x_{r}(t) · v^{+} = v^{+} for all r ∈ Φ^{+}. For all μ ∈ X with M_{λ}(μ) ≠ 0, it follows that μ ≤ λ.

Note that each field automorphism α of k̄ induces a group automorphism α of 𝔾 by defining α(x_{r}(t)) = x_{r}(α(t)), r ∈ Φ, t ∈ k̄.

The isomorphism classes of simple k̄𝔾-modules can now be described as follows:

Theorem 2.3 (Chevalley, Kostant, Steinberg, see [23, §2.1]). Let α_{i} be the field automorphism of k̄ which is defined by α_{i}(t) = t^{p^{i}}. Let λ ∈ X^{+}, such that λ ∈ X_{q} for a suitable q = p^{m}, λ = μ_{0} + pμ_{1} + ⋯ + p^{m−1}μ_{m−1} with μ_{i} ∈ X_{p}. Then

M_{λ} = M_{μ_{0}}^{α_{0}} ⊗k̄ M_{μ_{1}}^{α_{1}} ⊗k̄ ⋯ ⊗k̄ M_{μ_{m−1}}^{α_{m−1}}

is a simple k̄𝔾–module. All M_{λ} (λ ∈ X^{+}) are pairwise non-isomorphic and exhaust the isomorphism classes of simple k̄𝔾-modules.

Let q = p^{n} and let F = GF(q) be the finite subfield of k̄ with q elements. Then the finite Chevalley group G = 𝔾(F) over F is generated by {x_{r}(t) | r ∈ Φ, t ∈ F } with the same relations as 𝔾. Note: Ḡ = G/center(G) is a simple group except for the cases SL(2, 2), PSL(2, 3), Sp(4, 2) and G_{2}(2) [7, Thm. 11.1.2].

The twisted groups are certain subgroups of finite Chevalley groups: Suppose G is a finite Chevalley group of type A_{l}, l ≥ 2, D_{l}, l ≥ 4, E_{6},

or D_{4}, respectively, over a finite field F. Let ω be a nontrivial symmetry of order e of the Dynkin diagram and let γ be the corresponding graph automorphism. Let α be a nontrivial field automorphism such that Ω = γα satisfies Ω^{e} = 1, and let F_{0} be the subfield of F fixed by α. Note that F is of the form F = GF(p^{ne}) and F_{0} = GF(p^{n}). If G^{1} is defined to be the group of all elements of G fixed by Ω, then G^{1} is a twisted group of type ^{2}A_{l}, l ≥ 2, 2D_{l}, l ≥ 4, ^{2}E_{6}, or ^{3}D_{4}, respectively. Ḡ^{1} = G^{1}/center(G^{1}) is simple except for PSU(3, 2^{2}) [7, Thm. 14.4.1].

The definition of the finite Suzuki and Ree groups is slightly different: Suppose G is a finite Chevalley group of type B_{2}, G_{2}, or F_{4}, respectively,

over a finite field F which has characteristic 2 for B_{2} and F_{4}, and characteristic 3 for G_{2}. Let ω be the nontrivial symmetry of the Dynkin diagram with corresponding graph automorphism γ. Let α be a nontrivial field automorphism such that Ω^{2} = 1 for Ω = γα. Note that F is of the form F = GF(p^{2m+1}), p = 2 or 3. If G^{1} is the group of all elements of G fixed by Ω, then G^{1} is a Suzuki group if G has type B_{2}, and a Ree group if G has type G_{2} or F_{4}. G^{1} is simple for all positive m [7, Thm. 14.4.1].

Since the k̄𝔾-modules M_{λ}, λ ∈ X_{q}, define by restriction k̄G-modules, respectively k̄G^{1}-modules, we get the following theorem:

Theorem 2.4 ([43, Thm. 7.4, Thm. 9.3 and Thm. 12.2]). Let G be a finite Chevalley group over GF(p^{n}), or a twisted group of type ^{2}A_{l}, l ≥ 2, 2D_{l}, l ≥ 4, ^{2}E_{6}, or ^{3}D_{4} over GF(p^{ne}) where e is the order of the symmetry of the corresponding Dynkin diagram. Then every simple k̄G–module can be expressed uniquely as a tensor product

M_{0}^{α_{0}} ⊗k̄ M_{1}^{α_{1}} ⊗k̄ ⋯ ⊗k̄ M_{n−1}^{α_{n−1}} with M_{i} ∈ M.

In case that G is a Suzuki or Ree group of type ^{2}B_{2}, ^{2}G_{2}, or ^{2}F_{4} over GF(p^{n}), p = 2 or 3, set X′_{p} = {λ = ∑ c_{i}λ_{i} ∈ X_{p} | c_{i} = 0 for all long roots s_{i} ∈ Δ} and M′ = {M_{λ} | λ ∈ X′_{p}}. Then every simple k̄G–module can be expressed uniquely as a tensor product

M_{0}^{α_{0}} ⊗k̄ M_{1}^{α_{1}} ⊗k̄ ⋯ ⊗k̄ M_{n−1}^{α_{n−1}} with M_{i} ∈ M′.

Remark 2.5. Let G be a finite Chevalley group, a twisted group or a finite Suzuki or Ree group.

(i) Since the simple k̄G-modules are given as restrictions of certain simple modules of the corresponding Chevalley group 𝔾 of universal type, we can attach weights λ ∈ X^{+} to the simple k̄G-modules. If we choose λ ∈ X_{p^{n}}, n as in Theorem 2.4, then these weights determine the simple k̄G-modules uniquely up to isomorphisms. In this sense we will write the simple k̄G-modules also as M_{λ} for suitable weights λ ∈ X_{p^{n}}. Furthermore, the Brauer character corresponding to M_{λ} will be denoted by β_{λ}.

(ii) We will also use M_{λ} to denote the k̄G–module obtained by restriction of the k̄𝔾–module M_{λ} in case λ ∈ X^{+}\X_{q}. Note that then this restriction is not necessarily a simple k̄G–module. Again the corresponding Brauer character is denoted by β_{λ}.

(iii) The simple modules for the groups Ḡ = G/center(G) are exactly those

simple k̄G–modules on which the center acts trivially. (iv) For the finite groups the algebraically closed field k̄ can be replaced

by any splitting field of the same characteristic.

We want now to describe some general properties of the considered finite groups of Lie type which are also important for the proofs of the Theorems 1 and 2.

For the remainder of the paper, we fix the following notation:

Let G be a finite Chevalley group over GF(p^{n}), a twisted group over GF(p^{ne}), or a finite Suzuki or Ree group over GF(p^{n}) as in Theorem 2.4, and let q = p^{n}. Let Z = center(G) and let Ḡ = G/Z. The group automorphism of G induced by the field automorphism t ↦ t^{p^{i}} will be denoted by α_{i}. Then ᾱ_{i} with ᾱ_{i}(gZ) = α_{i}(g)Z is the corresponding group automorphism of Ḡ. Let σ ∈ Aut_{n}(ℤG) and σ̄ ∈ Aut_{n}(ℤḠ) be arbitrary elements.

Let (K, R, k) be a p-modular system with K sufficiently large relative to G such that k̄ is an algebraic closure of k. Since k is a splitting field for G, the isomorphism classes of simple kG-modules can be identified with those of simple k̄G-modules. Let 𝔾 be the Chevalley group of universal type over k̄ corresponding to G.

Lemma 2.6. The action of σ on the simple kG-modules S commutes with the action of α_{i} for all i, i.e. S^{σα_{i}} ≅ S^{α_{i}σ} for all i. The action of α_{i} on the p-regular conjugacy classes of G is given by Cl(g) ↦ Cl(g^{p^{i}}).

Proof. Because of Proposition 2.1 and Theorem 2.4, it suffices to show this for the simple modules M ∈ M or M′, respectively. Consider M as a k̄G-module. Thus we have to show M^{α_{i}σ} ≅ M^{σα_{i}} as k̄G-modules.

Suppose ρ : G → GL(m, k̄) is the representation corresponding to M as a k̄G-module. Let ν_{i} be the automorphism of GL(m, k̄), which maps a matrix (a_{rs}) to (a_{rs}^{p^{i}}). Then, if M^{(i)} denotes the module corresponding to the representation ν_{i} ∘ρ, we get M^{α_{i}} ≅ M^{(i)} as k̄G-modules (see for example [31, Prop. 5.4.2.(i)]). A simple matrix calculation shows that, if ξ_{1}, …, ξ_{m} are the eigenvalues of ρ(g), then ξ_{1}^{p^{i}}, …, ξ_{m}^{p^{i}} are the eigenvalues of ν_{i}(ρ(g)) and of ρ(g^{p^{i}}).

If φ is the Brauer character corresponding to ρ, and φ^{(i)} is the Brauer character corresponding to ν_{i} ∘ ρ, then it follows that

φ^{α_{i}}(g) = φ^{(i)}(g) = φ(g^{p^{i}}) (2.1)

for all p-regular elements g ∈ G.

Since Equation (2.1) is valid for all Brauer characters of G by Proposition 2.1, and since g ↦ g^{p^{i}} is a permutation of the p-regular elements, the permutation of the p-regular conjugacy classes of G corresponding to α_{i} is given by Cl(g) ↦ Cl(g^{p^{i}}).

Since σ preserves the power map, i.e. (C^{a})^{σ} = (C^{σ})^{a} for all conjugacy classes C = Cl(g) and C^{a} = Cl(g^{a}), it follows that φ^{α_{i}σ} = φ^{σα_{i}} and therefore, M^{α_{i}σ} ≅ M^{σα_{i}} for all M ∈ M or M′, respectively. □

Corollary 2.7. The operation of ᾱ_{i} on the p-regular conjugacy classes of Ḡ is given by Cl(ḡ) ↦ Cl(ḡ^{p^{i}}). In particular, it follows that S^{σ̄ᾱ_{i}} ≅ S^{ᾱ_{i}σ̄} for all simple kḠ-modules S.

Proof. By Lemma 2.6, α_{i}(Cl(g)) = Cl(g^{p^{i}}), i.e. α_{i}(g) is conjugate to g^{p^{i}} in G for all p-regular elements g ∈ G. Therefore, ᾱ_{i}(gZ) is conjugate to (gZ)^{p^{i}} in Ḡ, i.e. ᾱ_{i}(ḡ) is conjugate to ḡ^{p^{i}} for all p-regular elements ḡ ∈ Ḡ. Thus we have that ᾱ_{i}(Cl(ḡ)) = Cl(ḡ^{p^{i}}). Because ḡ ↦ ḡ^{p^{i}} is a permutation of the p-regular elements of Ḡ and because σ̄ preserves the power map, the corollary follows. □

With respect to the weights of tensor products we have the following result.

Lemma 2.8. Let λ, μ ∈ X^{+}, and let M_{λ} and M_{μ} be the two (not necessarily simple) kG-modules which are restrictions of the corresponding simple k̄𝔾modules. Then M_{λ+μ} is a factor module of M_{λ} ⊗ M_{μ} as kG-modules. Furthermore, as k̄𝔾-modules, λ+μ is the highest weight occurring in M_{λ} ⊗M_{μ}.

Suppose now that M_{λ} and M_{μ} are simple kG-modules with λ, μ ∈ X_{q}. If M_{λ}⊗M_{μ} is also a simple kG-module then M_{λ}⊗M_{μ} ≅ M_{λ+μ} as kG-modules. Since λ + μ ∈ X_{pq}, it follows that λ + μ = ν_{0} + qν_{1} for appropriate ν_{0} ∈ X_{q} and ν_{1} ∈ X_{p}. Thus

M_{λ+μ} ≅ M_{ν_{0}} ⊗ M_{ν_{1}}^{α_{n}}

as kG-modules. Note that M_{ν_{1}}^{α_{n}} ≅ M_{ν_{1}} if G is a finite Chevalley group or a Suzuki or Ree group. Otherwise M_{ν_{1}}^{α_{n}} ≅ M_{ω^{−1}(ν_{1})} where ω is the symmetry of the Dynkin diagram in the definition of the twisted groups.

Proof. Let λ, μ ∈ X^{+}. Then there exist maximal vectors v, respectively w, of highest weight λ, respectively μ, in the k̄𝔾-module M_{λ}, respectively M_{μ}. By [22, §31.4], v ⊗ w is then a maximal vector of the k̄𝔾-module M_{λ} ⊗ M_{μ} of weight λ+μ, which is the highest weight occurring in M_{λ} ⊗M_{μ}. So M_{λ+μ} is a factor module of M_{λ} ⊗ M_{μ} as k̄𝔾-modules and thus as kG-modules.

Let now λ, μ ∈ X_{q}. If M_{λ} ⊗ M_{μ} is simple as a kG-module then M_{λ} ⊗ M_{μ} is also simple as a k̄𝔾-module. Thus M_{λ} ⊗ M_{μ} is simple with weight λ + μ. Since λ, μ ∈ X_{q} and 2(q − 1) ≤ pq − 1, it follows that λ + μ = ν_{0} + qν_{1} for appropriate ν_{0} ∈ X_{q} and ν_{1} ∈ X_{p}. Thus M_{λ+μ} ≅ M_{ν_{0}} ⊗_{k} M_{ν_{1}}^{α_{n}} as kG-modules. □

We want now to outline the strategy to prove the conjecture (ZC) for certain G and Ḡ.

Definition 2.9. Denote by X̄^{+} all dominant weights λ ∈ X^{+} such that M_{λ} is a kḠ-module. In case that G is a Suzuki or a Ree group, let X^{+}′ = {λ = ∑ c_{i}λ_{i} ∈ X^{+} | c_{i} = 0 for all long roots s_{i} ∈ Δ}. Let Y^{+} be either X^{+}, X^{+}′, or X̄^{+}.

(i) A relation ≺ on Y^{+} is called a weak order if ≺ is reflexive and transitive.

(ii) An allowable system on Y^{+} is a subset R ⊆ Y^{+} such that all λ ∈ Y^{+} can be written as

λ = ∑_{ρ∈R} d_{ρ}ρ

for a unique choice of integers d_{ρ} ≥ 0.

(iii) Let R be an allowable system and ≺ a weak order on Y^{+}. Then ≺ is called an allowable weak order if the following properties are fulfilled:

(a) If λ ∈ {0} ∪ R and μ ∈ Y^{+} with μ ≺ λ, then μ ∈ {0} ∪ R.

(b) If μ_{1} ≤ μ_{2} and μ_{1} ≠ μ_{2}, it follows that μ_{1} ≺ μ_{2} and μ_{2} ⊀ μ_{1}.

(c) If λ = ∑ d_{ρ}ρ is a dominant weight with λ ∉ {0} ∪ R, then there exists ρ_{0} ∈ R with ρ_{0} ≺ λ and λ − ρ_{0} ≺ λ, but λ ⊀ λ − ρ_{0}.

Remark 2.10. (i) If ≺ is a weak order on Y^{+} then we can define an equivalence relation ∼ on Y^{+} by

λ ∼ μ ⇔ λ ≺ μ and μ ≺ λ.

Then ≺ defines a partial ordering of Y^{+}/ ∼, the equivalence classes of Y^{+} with respect to ∼.

(ii) For Y^{+} = X^{+} (respectively X^{+}′) we can and will always choose the

fundamental weights λ_{i} ∈ X_{p} (respectively X′_{p}) as allowable system. (iii) Definition 2.9 is written in this generality since we want to prove (ZC)

for groups G and Ḡ. (iv) The order relation ≤ on X^{+} does not always fulfill property (c). For

example, for SL(3, q), λ_{1} ≰ λ_{1} + λ_{2} and λ_{2} ≰ λ_{1} + λ_{2}.

The following proposition plays an important role in the examination of (ZC).

Proposition 2.11. Let (H, Y^{+}) be either (G, X^{+}), (G, X^{+}′), or (Ḡ, X̄^{+}), let τ be either σ or σ̄, and let R be an allowable system of Y^{+}. Suppose

(i) there exists a group automorphism α ∈ Aut(H) with M_{ρ}^{τ} ≅ M_{ρ}^{α} for all ρ ∈ R, and

(ii) there exists an allowable weak order ≺ on Y^{+}. Then τ and α induce the same operation on the p-modular character table of H.

Proof. We have to show that β_{λ}^{τ} = β_{λ}^{α} for all λ ∈ Y^{+} ∩ X_{q}.

By (i) there exists a group automorphism α such that β_{ρ}^{τ} = β_{ρ}^{α} for all ρ ∈ R. Thus, by property (a) of ≺, the statement is valid for the smallest weights with respect to ≺.

Let now λ ∈ Y^{+}, λ ∉ {0} ∪ R. Suppose β_{μ}^{τ} = β_{μ}^{α} for all μ ∈ Y^{+} with μ ≺ λ and λ ⊀ μ, which means, by property (b) of ≺, in particular for all μ ∈ Y^{+} with μ ≤ λ, μ ≠ λ. By property (c), there exists ρ_{0} ∈ R such that μ_{1} = ρ_{0} ≺ λ, and μ_{2} = λ − ρ_{0} ≺ λ, but λ ⊀ μ_{2}. Thus β_{μ_{s}}^{τ} = β_{μ_{s}}^{α} for s ∈ {1, 2}. By Lemma 2.8, β_{μ_{1}} ⊗ β_{μ_{2}} can be written as

β_{μ_{1}} ⊗ β_{μ_{2}} = ∑_{ν∈Y^{+}} a_{ν}β_{ν} with a_{ν} ≥ 0

such that a_{λ} ≠ 0 and ν ≤ λ for all other ν ∈ Y^{+} with a_{ν} ≠ 0. Using Proposition 2.1, it follows that β_{λ}^{τ} = β_{λ}^{α}. □

Remark 2.12. (i) Proposition 2.11 implies that τ operates as α on all ordinary irreducible characters which are not p-exceptional. Note that a character χ ∈ Irr(H) is called p-exceptional, if there exists another character χ′ ∈ Irr(H) such that χ and χ′ have the same values on the p-regular classes of H.

(ii) All group automorphisms of G, respectively Ḡ, are given as the composition of an inner, a diagonal, a graph and a field automorphism [7, Thm. 12.5.1]. Since the inner and diagonal automorphisms operate trivially on the p-regular classes, α can be chosen as the composition of a graph and a field automorphism.

Let H, τ and α be as in Proposition 2.11. Then τα^{−1} operates trivially on all ordinary irreducible characters which are not p-exceptional. We will use ad-hoc techniques to refine α to a group automorphism β of H such that τβ^{−1} fixes also all p-exceptional characters. Using Lemma 2.2 this then implies that (ZC) is valid for H.

In this context, l-blocks of H which have cyclic defect groups and their corresponding Brauer trees are important. Note that usually the prime l is different from p.

There are two interpretations of the Brauer tree corresponding to a block B with cyclic defect groups. One approach is to define the Brauer tree by only using the category of finitely generated B-modules. This is described in [1, Chapter V]. The other possibility is to define the Brauer tree using the decomposition map. This follows from Brauer’s and Dade’s theory of blocks with cyclic defect groups (see [12]). By [12, Thm. 1] it follows that these two definitions coincide. As a reference for both interpretations of the Brauer tree of B use for example [6].

With respect to the operation of τ on the Brauer trees corresponding to l-blocks with cyclic defect groups, we need the following two results. Note that τ permutes the l-blocks of H.

Lemma 2.13. Let B be an l-block of H with cyclic defect groups such that τ fixes B. Then τ induces a graph automorphism of the Brauer tree of B which is either trivial or a rotation.

Proof. This follows from [6, Prop. 3.7] and the proof of [6, Cor. 3.9]. □

Lemma 2.14 ([6, Cor. 3.10]). Let H have cyclic Sylow l-subgroups and let B be the principal l-block. If χ ∈ Irr(H) belongs to B and if χ is not lexceptional, then τ fixes χ.

We use these two lemmas in the following way. To refine α of Proposition 2.11, we want to show that τα^{−1} operates on the p-exceptional characters as the power of a diagonal automorphism. In case all diagonal automorphisms of H are inner, we use Lemma 2.14 for certain l to show that τα^{−1} fixes all p-exceptional characters. In the case of the Ree groups of type F_{4}, Lemma 2.13 is used to eliminate operations on characters which are not induced by group automorphisms.

In case that H has a nontrivial diagonal automorphism δ, the characters permuted by δ have to be examined separately. Here we use, if necessary, Lemma 2.13 to study all possible operations on Brauer trees which contain such characters.

Remark 2.15. (i) To determine the p-exceptional characters we use mostly the generic character tables provided in CHEVIE [17]. As outlined in the introduction there are literature references for most of the considered tables, which are either verified or corrected in CHEVIE.

(ii) In the examination of the p-exceptional characters, these characters are decomposed into a union of pairwise disjoint subsets such that all characters which have the same values on the p-regular classes are in one such subset.

(iii) When Brauer trees are given, a black circle • denotes the exceptional vertex.

According to the above described strategy, the proofs of the validity of (ZC) for certain G (respectively Ḡ) will always be divided into two parts:

(A) Operation of σ (respectively σ̄) on the p-modular character table.

Here we only have to check the hypotheses of Proposition 2.11. (B) Operation of σ (respectively σ̄) on the p-exceptional characters.

Here we use ad-hoc arguments together with Lemmas 2.13 and 2.14.

The proofs involve the following Dynkin diagrams: A_{2}, B_{2}, G_{2}, D_{4}, F_{4}.

[figure: A_{2}:; λ_{1}; λ_{2}; B_{2}:; >; λ_{1}; λ_{2}; G_{2}:; >; λ_{1}; λ_{2}; D_{4}:; λ_{1}; λ_{2}; λ_{4}; λ_{3}; F_{4}:; >; λ_{1}; λ_{2}; λ_{3}; λ_{4}]

The arrow always points to the shorter roots.

A weight λ = ∑ c_{i}λ_{i} will be denoted by λ = c_{1} ⋯ c_{l}.

3. The Ree groups ^{2}G_{2}(q^{2}) and ^{2}F_{4}(q^{2}).

In this section, we want to prove that (ZC) is valid for the finite Ree groups of type G_{2} and F_{4}, and thus establish Theorem 1.

The Ree groups ^{2}G_{2}(q^{2}) and ^{2}F_{4}(q^{2}) are simple except for ^{2}G_{2}(3) and 2F_{4}(2); their automorphisms are given by inner and field automorphisms. The generic ordinary character table of ^{2}G_{2}(3^{2m+1}) was mostly determined by Ward in [44]. Shinoda described the conjugacy classes of ^{2}F_{4}(2^{2m+1}) in [40]. The complete ordinary character table of ^{2}G_{2}(3^{2m+1}) can be found in CHEVIE; most of the character table of ^{2}F_{4}(2^{2m+1}) and all Green functions are also listed there.

Proposition 3.1. (i) The conjecture (ZC) is valid for ^{2}G_{2}(q^{2}) for all

q^{2} = 3^{2m+1}, m ≥ 1. (ii) The conjecture (ZC) is valid for ^{2}F_{4}(q^{2}) for all q^{2} = 2^{2m+1}, m ≥ 1.

Proof. Let G = ^{2}G_{2}(q^{2}) or G = ^{2}F_{4}(q^{2}), respectively.

Part (A).

(i) X′_{p} = {00, 01, 02} with dim(M_{01}) = 7 and dim(M_{02}) = 27. Because of dimensions, there exists 0 ≤ i ≤ 2m with M_{01}^{σ} ≅ M_{01}^{α_{i}}. Since ≤ defines an allowable weak order ≺ on X^{+}, the hypotheses of Proposition 2.11 are satisfied.

(ii) X′_{p} = {0000, 0010, 0001, 0011} with dim(M_{0001}) = 26, dim(M_{0010}) = 246 and dim(M_{0011}) = 4096. Because of dimensions, there exist 0 ≤ i, j ≤ 2m with M_{0001}^{σ} ≅ M_{0001}^{α_{i}} and M_{0010}^{σ} ≅ M_{0010}^{α_{j}}. We have to show that i = j. If i ≠ j then σ maps the simple module M_{0001}^{α_{2m+1−i}} ⊗ M_{0010}^{α_{2m+1−j}} to M_{0001} ⊗ M_{0010}. This implies that M_{0001} ⊗ M_{0010} must be simple, thus by Lemma 2.8 of weight 0011. A comparison of the dimensions shows that this is impossible. Thus it follows that i = j which yields hypothesis (i) of Proposition 2.11. ≤ defines again an allowable weak order ≺ on X^{+}, which is hypothesis (ii).

Part (B).

(i) G does not have any nontrivial diagonal automorphisms, thus we have to show that σα_{i}^{−1}, α_{i} from Part (A), fixes all ordinary characters which are p-exceptional. The generic character table given in CHEVIE shows that there are exactly 6 unipotent characters which are p-exceptional: Using the notation of [20], these characters correspond to {ξ_{3}, ξ_{4}}, {ξ_{5}, ξ_{6}} and {ξ_{7}, ξ_{8}}. If l > 3 is a rational prime which divides |G|, then G has cyclic Sylow l-subgroups [25, XI Thm. 13.2]. By [20, Thm. 4.2], ξ_{5}, ξ_{6} and ξ_{7}, ξ_{8} lie in the principal l-block for l|(q^{2} + 1) and are not l-exceptional. By [20, Thm. 4.3], ξ_{3}, ξ_{4} and ξ_{5}, ξ_{6} lie in the principal l-block for l|(q^{2} + \sqrt{3}q + 1) and are not lexceptional. Thus, by Lemma 2.14, σα_{i}^{−1} fixes all ordinary characters, which proves Proposition 3.1(i).

(ii) Since the generic character table of G given in CHEVIE is not complete, we have to look at this case more carefully. Again we have to show that σα_{i}^{−1} operates trivially on the characters which are pexceptional.

Let 𝔾 be the Chevalley group of universal type of type F_{4} over k̄, and let F be the Frobenius map (notation used as in [8, §1.17]) with 𝔾^{F} = G. Using the Jordan decomposition for characters, the ordinary irreducible characters of G can be described as follows (cf. [8, §12.9]): Since the Ree groups are self-dual, the characters of G are parametrized by G-conjugacy classes of pairs (s, λ) where s is a semisimple element of G and λ is a unipotent character of C_{G}(s). This character is then denoted by χ_{s,λ}. Since G is self-dual, the G-conjugacy classes of semisimple elements of G correspond to geometric conjugacy classes.

We use the notation in [40] for the 11 conjugacy classes of the maximal tori and the representatives of the semisimple conjugacy classes, i.e. of the semisimple class types. Note that the G-conjugacy classes of the maximal tori 𝕋 of 𝔾 are uniquely determined by the isomorphism type of 𝕋^{F}.

In CHEVIE, all ordinary irreducible characters of G can be found except for those corresponding to the semisimple representatives t_{7}, t_{9}, t_{12} and t_{13}. From the part of the character table given in CHEVIE we can read off the following:

Exactly 10 of the unipotent characters are p-exceptional. These are, using the notation of [20], {ξ_{5}, ξ_{6}}, {ξ_{7}, ξ_{8}}, {ξ_{15}, ξ_{16}}, {ξ_{17}, ξ_{18}} and {ξ_{19}, ξ_{20}}. By [20, Thm. 4.6 and Thm. 4.7] and Lemma 2.14, σα_{i}^{−1} operates trivially on these 10 unipotent characters.

Looking at the other characters listed in CHEVIE, {χ_{23}(k), χ_{24}(k)} are p-exceptional for all parameters k. According to the Jordan decomposition, the characters χ_{22}(k) through χ_{25}(k) belong to the semisimple class type t_{1}. If s is the representative of the G-conjugacy class of type t_{1} corresponding to k, then C_{G}(s) ≅ ℤ_{q^{2}−1} ×^{2}B_{2}(q^{2}). Thus there are four possibilities for λ, namely the unipotent characters of ^{2}B_{2}(q^{2}): 1, μ, μ̄ and St. Therefore we have the correspondence χ_{22}(k) ↔ χ_{s,1}, χ_{23}(k) ↔ χ_{s,μ}, χ_{24}(k) ↔ χ_{s,μ̄} and χ_{25}(k) ↔ χ_{s,St}.

For a rational prime l|(q^{2} − \sqrt{2}q + 1), we have the following Brauer trees for such s [19, Satz D.3.9]:

[figure: χ_{s,1}; χ_{s,St}; χ_{s,μ}; χ_{s,μ̄}]

Thus, by Lemma 2.13, σα_{i}^{−1} cannot induce the permutation (χ_{s,μ}, χ_{s,μ̄}) which means that σα_{i}^{−1} cannot permute χ_{23}(k) and χ_{24}(k).

There are similar Brauer trees for the semisimple class types t_{7} and t_{9}. We have C_{G}(t_{7}) ≅ ℤ_{q^{2}−\sqrt{2}q+1} × ^{2}B_{2}(q^{2}), C_{G}(t_{9}) ≅ ℤ_{q^{2}+\sqrt{2}q+1} × 2B_{2}(q^{2}). For each G-conjugacy class of s of type t_{7} or t_{9}, there are again 4 characters χ_{s,1}, χ_{s,μ}, χ_{s,μ̄} and χ_{s,St}. The Brauer trees are then as follows:

For s of type t_{9} and l|(q^{2} − \sqrt{2}q + 1), we get a tree analogous to the one for s of type t_{1} [19, Satz D.3.9].

For s of type t_{7} and l|(q^{2} + \sqrt{2}q + 1), the Brauer trees look like [19, Satz D.3.10]:

[figure: χ_{s,1}; χ_{s,St}; χ_{s,μ}; χ_{s,μ̄}]

Thus, by Lemma 2.13, σα_{i}^{−1} cannot permute χ_{s,μ} and χ_{s,μ̄} for s of type t_{7} or t_{9}.

It remains to show that σα_{i}^{−1} operates trivially within each row of the ordinary generic character table.

This is clear for all the characters listed in CHEVIE except for those corresponding to the semisimple class type t_{1}. Thus we have to look at the characters corresponding to the semisimple types t_{1}, t_{7}, t_{9} and t_{12}, t_{13}.

Since t_{12} and t_{13} are regular, i.e. the corresponding characters are of the form ±R_{𝕋,θ} with 𝕋 of type T (6) for t_{12} and of type T (7) for t_{13} and suitable θ in general position, the claim follows almost immediately.

For s of type t = t_{1}, t_{7} or t_{9}, respectively, we look at χ_{s,1} + χ_{s,St}. If s and s′ belong to different G-conjugacy classes of type t, it follows that (χ_{s,1} + χ_{s,St})|_{G_{p′}} ≠ (χ_{s′,1} + χ_{s′,St})|_{G_{p′}}. Thus it follows that either

(χ_{s,1})|_{G_{p′}} ≠ (χ_{s′,1})|_{G_{p′}} or (χ_{s,St})|_{G_{p′}} ≠ (χ_{s′,St})|_{G_{p′}}. Therefore σα_{i}^{−1}

either does not permute χ_{s,1} and χ_{s′,1} or it does not permute χ_{s,St} and χ_{s′,St}. Using the above Brauer trees, χ_{s,1}, χ_{s,μ}, χ_{s,μ̄} and χ_{s,St} lie in the same Brauer tree of an l-block and are not l-exceptional. Thus σα_{i}^{−1} operates trivially on the rows of the generic ordinary character table corresponding to the semisimple class type t.

This proves Proposition 3.1(ii).

□

Remark 3.2. The conjecture (ZC) is also valid for the simple Tits group 2F_{4}(2)′, which is the normal subgroup of ^{2}F_{4}(2) of index 2. Since the proof uses different methods from those described here, we state this result without proof. Proof of Theorem 1. The simple groups with abelian Sylow 2-subgroups have been classified by Walter [25, XI Thm. 13.7]. These are PSL(2, 2^{f}) for f ≥ 1, PSL(2, q) for q = p^{f}, q ≠ 3, with q ≡ 3 (8) or q ≡ 5 (8), the Janko group J_{1} and the finite Ree groups of type G_{2}.

The validity of (ZC) for PSL(2, p^{f}) for arbitrary rational primes p has been proved in [4, Prop. 3.2 and Prop. 4.1]. The theory of principal blocks with cyclic defect groups together with Lemma 2.14 shows that (ZC) holds for J_{1}. Thus, together with Proposition 3.1(i), Theorem 1 follows.

4. Finite groups of Lie type of rank 2.

In this section we want to deal with finite Chevalley groups and twisted groups of rank 2. This will especially prove Theorem 2. 4.1. The linear and unitary groups SL(3, q), PSL(3, q), SU(3, q^{2}) and PSU(3, q^{2}). The groups SL(3, q) are simple for q ≢ 1 (3), and SU(3, q^{2}) are simple for q ≢ −1 (3). In these cases, the group automorphisms are generated by inner, graph and field automorphisms. In all other cases, there also exist diagonal automorphisms. The ordinary generic character tables have been determined in [41] and can also be found in CHEVIE. We first prove the following:

Proposition 4.1. The conjecture (ZC) is valid for SL(3, q) and SU(3, q^{2}) for all q = p^{m}, where p is a rational prime and m ≥ 1.

Proof. Let G = SL(3, q) or G = SU(3, q^{2}), respectively.

Part (A).

By [33, Thm. 1.1], all simple kG-modules with dimension smaller than or equal to 5 are of the form M_{10}^{α_{j}} or M_{01}^{α_{j}} for some j. Note that if γ is the nontrivial graph automorphism for G = SL(3, q), and if γ is the field automorphism α_{m} for G = SU(3, q^{2}), then M_{10}^{γ} ≅ M_{01}. Furthermore, since in case G = SL(3, q), γ operates on the conjugacy classes as (Cl(g))^{γ} = Cl(g^{−1}) and since σ preserves the power map, the action of σ on the pmodular Brauer characters commutes with the action of γ. This implies hypothesis (i) of Proposition 2.11. The weak order ≺ on X^{+} for hypothesis (ii) is defined as follows:

Let μ_{1} = b_{1}b_{2} and μ_{2} = c_{1}c_{2} be two dominant weights. Then μ_{1} ≺ μ_{2} if and only if b_{1} + b_{2} ≤ c_{1} + c_{2}. To show that ≺ is allowable, we only have to show that μ_{1} ≤ μ_{2}, μ_{1} ≠ μ_{2} implies μ_{1} ≺ μ_{2} and μ_{2} ⊀ μ_{1}. If s_{1} and s_{2} are the fundamental roots corresponding to 10 and 01, then we have

\begin{aligned}10 & = & \frac{1}{3} ( & 2 · s_{1} & + & s_{2}), \\ 01 & = & \frac{1}{3} ( & s_{1} & + & 2 · s_{2}). \end{aligned}

Thus it follows that μ_{1} ≤ μ_{2} if and only if

\begin{aligned}2b_{1} & + & b_{2} & ≤ & 2c_{1} & + & c_{2}, \\ b_{1} & + & 2b_{2} & ≤ & c_{1} & + & 2c_{2}, \end{aligned}

and 2(c_{1} − b_{1}) + (c_{2} − b_{2}) and (c_{1} − b_{1}) + 2(c_{2} − b_{2}) are multiples of 3. This implies that b_{1} +b_{2} ≤ c_{1} +c_{2}. If b_{1} +b_{2} = c_{1} +c_{2} and μ_{1} ≠ μ_{2}, then μ_{1} ≰ μ_{2}. Thus μ_{1} ≤ μ_{2}, μ_{1} ≠ μ_{2} implies μ_{1} ≺ μ_{2} and μ_{2} ⊀ μ_{1}.

Part (B).

G has only p-exceptional characters if q ≡ 1 (3) and G = SL(3, q) or if q ≡ −1 (3) and G = SU(3, q^{2}).

Let now G = SL(3, q) and q ≡ 1 (3). In the notation of CHEVIE, there are 9 characters which are p-exceptional: {χ_{6}, χ_{7}, χ_{8}}, {χ_{11}, χ_{12}, χ_{13}} and {χ_{14}, χ_{15}, χ_{16}}. The following actions on these characters are induced by group automorphisms: There exists a diagonal automorphism δ with operation δ̃ = (χ_{6}, χ_{7}, χ_{8})(χ_{11}, χ_{12}, χ_{13})(χ_{14}, χ_{15}, χ_{16}). Note that we use here cycle notation. For the graph automorphism γ we get as operation γ̃ = (χ_{11}, χ_{14})(χ_{12}, χ_{15})(χ_{13}, χ_{16}). If p ≡ 1 (3), we get for the field automorphism α_{1} the same operation as for γ. We want to show that the operations on the 9 p-exceptional characters generated by group automorphisms of G are the only operations that can be induced by σ. Note that if χ_{11} and χ_{14} are permuted then this induces a nontrivial operation on the p-regular classes. On the other hand δ operates trivially on the p-regular classes. Thus we only have to show that all operations on these 9 characters induced by σ are generated by π_{1} = (χ_{6}, χ_{7}, χ_{8})(χ_{11}, χ_{12}, χ_{13})(χ_{14}, χ_{15}, χ_{16}) and π_{2} = (χ_{11}, χ_{14})(χ_{12}, χ_{15})(χ_{13}, χ_{16}).

Note that |G| = q^{3}(q − 1)^{2}(q + 1)(q^{2} + q + 1). By [24, II Satz 7.3], G has a cyclic subgroup of order q^{2} + q + 1. If l ≠ 3 is a rational prime which divides q^{2} + q + 1, then all Sylow l-subgroups are cyclic because (q^{2} +q +1, q −1) = 1 and (q^{2} +q +1, q +1) = 1. The characters χ_{11} through χ_{16} are not l-exceptional and lie in the following Brauer trees:

[figure: χ_{11}; χ_{13}; χ_{12}; χ_{14}; χ_{16}; χ_{15}]

By Lemma 2.13, it follows that all operations on the characters χ_{11}, χ_{12}, χ_{13}, χ_{14}, χ_{15}, χ_{16} induced by σ are generated by π_{1} and π_{2}. Because of the corresponding operations of π_{1} and π_{2} on the conjugacy classes, this also follows for the characters χ_{6}, χ_{7}, χ_{8}.

For G = SU(3, q^{2}) and q ≡ −1 (3), the argumentation is similar. Here we use that SU(3, q^{2}) has a cyclic subgroup of order q^{2} − q + 1 [32, Satz 4.2].

This proves Proposition 4.1. □

For the groups PSL(3, q) and PSU(3, q^{2}) we get a similar result. Note that PSL(3, q) and PSU(3, q^{2}) are simple except for PSU(3, 2^{2}) ≅ (ℤ_{3} × ℤ_{3}).Q_{8}.

Proposition 4.2. The conjecture (ZC) is valid for PSL(3, q) and PSU(3, q^{2}) for all q = p^{m}, where p is a rational prime and m ≥ 1.

Proof. Let Ḡ = PSL(3, q) or Ḡ = PSU(3, q^{2}), respectively. Let γ̄ be the nontrivial graph automorphism if Ḡ = PSL(3, q), and γ̄ = ᾱ_{m} if Ḡ = PSU(3, q^{2}). Since we can use the same argumentation as in Proposition 4.1 to deal with the characters which are p-exceptional, we only have to look at:

Part (A).

The modules M_{λ}, λ = a_{1}a_{2} ∈ X_{q}, are simple kḠ-modules if and only if a_{1}+2a_{2} ≡ 0 (3). This follows by looking at the center of SL(3, q) or SU(3, q^{2}) as described in [7, §12.1].

Let now λ = a_{1}a_{2} with a_{1} + 2a_{2} ≡ 0 (3). Then λ = a_{1} · 11 + \frac{1}{3}(a_{2} − a_{1}) · 03 if a_{1} ≤ a_{2} and λ = a_{2} · 11 + \frac{1}{3}(a_{1} − a_{2}) · 30 if a_{2} ≤ a_{1}. Note that |a_{2} − a_{1}| = \frac{1}{2}|a_{1} + 2a_{2} − 3a_{1}| is divisible by 3. We have to show that there exists α ∈ Aut(Ḡ) with M σ̄ ; 11 ≅ M_{11}^{α} and M σ̄ ; 03 ≅ M_{03}^{α}. This then implies hypothesis (i) of Proposition 2.11. For hypothesis (ii) we can use the same weak order ≺ as for SL(3, q) and SU(3, q^{2}).

By [2, Thm. 1], all modules of dimension 8 are of the form M ᾱ_{i} ; 11 for some 0 ≤ i ≤ m − 1 and, for p > 3, all modules of dimension 10 are of the form M_{03}^{γ̄^{ν} ᾱ_{j}} for some ν ∈ {0, 1} and some 0 ≤ j ≤ m − 1.

Let now p > 3. Then we get that M σ̄ ; 11 ≅ M ᾱ_{i} ; 11 and M σ̄ ; 03 ≅ M_{03}^{γ̄^{ν} ᾱ_{j}}. Note that M γ̄ ; 11 ≅ M_{11}. Thus we have to show that i = j. This follows, similarly as in the case of ^{2}F_{4}(q^{2}), from the fact that M_{11} ⊗ M_{03} and M_{11} ⊗ M_{30} are not simple. Note that they have both dimension 80 whereas M_{14} and M_{41} have both dimension ≤ 35 by Weyl’s formula (see [21, §24.3]). Thus we get the desired operation of σ̄ on these modules.

For p = 2, the situation is slightly more complicated. Again M σ̄ ; 11 ≅ M ᾱ_{i} ; 11 for some i. But in this case M_{03} ≅ M_{01}⊗M ᾱ_{1} ; 01 and has dimension 9. We have only to show that M σ̄ ; 03 ≅ M_{03}^{γ̄^{ν} ᾱ_{j}} for some ν and some j. The argumentation that i = j is then similar to the case p > 3.

The modules of dimension 9 have the form M_{01}^{γ̄^{ν_{1}} ᾱ_{i_{1}}} ⊗M_{01}^{γ̄^{ν_{2}} ᾱ_{i_{2}}} with ν_{1}, ν_{2} ∈ {0, 1}, and 0 ≤ i_{1} < i_{2} ≤ m − 1. This is a kḠ-module in case that i_{2} − i_{1} is odd for ν_{1} = ν_{2} and i_{2} − i_{1} is even for ν_{1} ≠ ν_{2}. For the twist of M_{03} under σ̄ we have to show that either i_{2} = i_{1} + 1 and ν_{1} = ν_{2} or (i_{1}, i_{2}) = (0, m − 1) and (ν_{1} = ν_{2} if Ḡ = PSL(3, q) or ν_{1} ≠ ν_{2} if Ḡ = PSU(3, q^{2})). Note that for m = 2, M_{03} and M_{30} are the only kḠ-modules of dimension 9. So let m ≥ 3.

Suppose first that ν_{1} ≠ ν_{2} and, if Ḡ = PSU(3, q^{2}), that (i_{1}, i_{2}) ≠ (0, m−1). If i_{2} = i_{1} + 1 then i_{2} − i_{1} is odd, thus the module is not a kḠ-module. So i_{2} ≠ i_{1} + 1. Then there exists s ∈ {i_{1} − i, i_{2} − i} such that s ≢ 0 (m) and

s ≢ 1 (m). Thus M_{03} ⊗ M ᾱ_{s} ; 11 ≅ M_{01} ⊗ M ᾱ_{1} ; 01 ⊗ M ᾱ_{s} ; 11 is a simple module which

would be mapped by σ̄ to M_{01}^{γ̄^{ν_{1}} ᾱ_{i_{1}}} ⊗ M_{01}^{γ̄^{ν_{2}} ᾱ_{i_{2}}} ⊗ M ᾱ_{i+s} ; 11. But this module is not simple, because neither M_{01} ⊗ M_{11} nor M_{10} ⊗ M_{11} are simple.

Suppose now that ν_{1} = ν_{2} and i_{2} ≠ i_{1} + 1 and, if Ḡ = PSL(3, q^{2}), that (i_{1}, i_{2}) ≠ (0, m−1). Again there exists s ∈ {i_{1} −i, i_{2} −i} such that s ≢ 0 (m) and s ≢ 1 (m). So we can argue in the same way as in the first case.

Therefore we get for M σ̄ ; 03 ≅ M_{01}^{γ̄^{ν_{1}} ᾱ_{i_{1}}} ⊗ M_{01}^{γ̄^{ν_{2}} ᾱ_{i_{2}}} that either i_{2} = i_{1} + 1 and ν_{1} = ν_{2} or (i_{1}, i_{2}) = (0, m − 1) and (ν_{1} = ν_{2} if Ḡ = PSL(3, q) or ν_{1} ≠ ν_{2} if Ḡ = PSU(3, q^{2})).

This proves Proposition 4.2. □

4.2. The symplectic groups Sp(4, q) and PSp(4, q). After the linear and unitary groups of type A_{2}, we consider now the symplectic groups Sp(4, q) and PSp(4, q). Note that Sp(4, 2^{m}) is simple for m ≥ 2, and Sp(4, 2) is isomorphic to the symmetric group on 6 elements. For even q, the group automorphisms are generated by inner, graph and field automorphisms. In all other cases, there exist also diagonal automorphisms, but no nontrivial graph automorphisms. The ordinary generic character table for even q has been determined in [14] and appears also in CHEVIE. For odd q, the generic character table was first established by Srinivasan [42], Przygocki [34] found some small mistakes; a preliminary version computed by Lübeck in CHEVIE shows that the following corrections to Przygocki’s version have to be made: For the characters ξ′_{41}, (1 − q)^{2} must be replaced by (1 − q^{2}) on the classes D_{1}, for ϕ_{3} the sign must be changed on C_{21}, and for ϕ_{7} the sign has to be changed on B_{3}.

We first prove the following proposition:

Proposition 4.3. The conjecture (ZC) is valid for Sp(4, q) for all q = p^{m} with p a rational prime and m ≥ 1.

Proof. Let G = Sp(4, q).

Part (A).

If p = 2, then X_{p} = {00, 01, 10, 11} with dim(M_{01}) = 4 = dim(M_{10}) and M_{11} ≅ M_{01} ⊗ M_{10}. If γ denotes the nontrivial graph automorphism of G, then M_{10} ≅ M_{01}^{γ}. By [10, Thm. (3.4)] it follows that

β_{01} ⊗ β_{01} = 4 · 1 + 2 · β_{10} + β_{01}^{α_{1}}. (4.1)

Because of dimensions, β_{01}^{σ} = β_{01}^{γ^{ν}α_{i}} for some ν ∈ {0, 1}, 0 ≤ i ≤ m − 1. Thus we get with Equation (4.1)

4 · 1 + 2 · β_{10}^{γ^{ν}α_{i}} + β_{01}^{γ^{ν}α_{i+1}} = 4 · 1 + 2 · β_{10}^{σ} + β_{01}^{σα_{1}}

and thus β_{10}^{σ} = β_{10}^{γ^{ν}α_{i}}. So hypothesis (i) of Proposition 2.11 follows for the case p = 2.

Now let p ≠ 2. Then dim(M_{01}) = 4 and dim(M_{10}) = 5. If M is a simple kG-module with 1 < dim(M) < 8 then there exists an automorphism α of G, such that either M ≅ M_{01}^{α} or M ≅ M_{10}^{α} [33, Thm. 1.1]. This implies hypothesis (i) in case p ≠ 2.

The weak order ≺ for hypothesis (ii) is defined as follows:

Let μ_{1} = b_{1}b_{2} and μ_{2} = c_{1}c_{2} be two different dominant weights. Then μ_{1} ≺ μ_{2} if and only if b_{1} + b_{2} < c_{1} + c_{2}, or b_{1} + b_{2} = c_{1} + c_{2} and b_{1} < c_{1}. Similarly to SL(3, q), it follows that μ_{1} ≤ μ_{2}, μ_{1} ≠ μ_{2} implies μ_{1} ≺ μ_{2} and μ_{2} ⊀ μ_{1}.

Part (B).

When p = 2, G has no p-exceptional characters. Therefore let p ≠ 2. We use here the notation given in [42] and [34]. The characters which

are p-exceptional are exactly the following pairs: {ξ_{21}(k), ξ_{22}(k)} for each k, {ξ′_{21}(k), ξ′_{22}(k)} for each k, {ξ_{41}(k), ξ_{42}(k)} for each k, {ξ′_{41}(k), ξ′_{42}(k)} for each k, {Φ_{1}, Φ_{2}}, {Φ_{3}, Φ_{4}}, {Φ_{5}, Φ_{6}}, {Φ_{7}, Φ_{8}}, {θ_{1}, θ_{2}}, {θ_{3}, θ_{4}}, {θ_{5}, θ_{6}} and {θ_{7}, θ_{8}}. The nontrivial diagonal automorphism δ of G permutes simultaneously the two characters of each such pair. Looking at the corresponding conjugacy classes it follows that the only possible nontrivial operation induced by σ on these pairs is the simultaneous permutation of each pair.

This proves Proposition 4.3. □

For the groups PSp(4, q) we get a similar result:

Proposition 4.4. The conjecture (ZC) is valid for PSp(4, q) for all q = p^{m} with p a rational prime and m ≥ 1.

Proof. Let Ḡ = PSp(4, q). Because of Proposition 4.3, we only have to show Proposition 4.4 for p ≥ 3. So let p ≥ 3. Since we can use the same argumentation as in Proposition 4.3 to deal with the characters which are exceptional for p, we only have to look at:

Part (A).

The modules M_{λ}, λ = a_{1}a_{2} ∈ X_{q}, are simple kḠ-modules if and only if a_{2} ≡ 0 (2). This follows by looking at the center of Sp(4, q) as described in [7, §12.1].

Let now λ = a_{1}λ_{1}+a_{2}λ_{2} = a_{1}a_{2} with a_{2} ≡ 0 (2). Then λ = a_{1}·10+ \frac{1}{2}a_{2}·02. We have to show that there exists α ∈ Aut(Ḡ) with M σ̄ ; 10 ≅ M_{10}^{α} and M σ̄ ; 20 ≅ M_{20}^{α}. This then implies hypothesis (i) of Proposition 2.11. For hypothesis (ii) we can use the same weak order ≺ as for Sp(4, q).

By [2, Thm. 1], all modules of dimension 5 are of the form M ᾱ_{i} ; 10 for some i, and all modules of dimension 10 are of the form M ᾱ_{j} ; 02 for some j. Thus we get the desired operation of σ̄ on these modules.

This proves Proposition 4.4. □

4.3. The finite Chevalley groups G_{2}(q). After the classical groups of type A_{2} and B_{2}, we want now to examine the exceptional groups of type G_{2}. The ordinary generic character tables of these groups can be found in [15] and in [9, 16] and have also been determined in CHEVIE.

G_{2}(q) is simple except for G_{2}(2) ≅ SU(3, 3^{2}).2 = Aut(SU(3, 3^{2})). For 3 |/ q, the group automorphisms of G_{2}(q) are generated by inner and field automorphisms. For q = 3^{m} there exists also a nontrivial graph automorphism of order 2.

We want to prove the following proposition:

Proposition 4.5. The conjecture (ZC) is valid for G_{2}(q) for all q = p^{m} with p a rational prime and m ≥ 1.

Proof. Let G = G_{2}(q).

Part (A).

If p = 3, then X_{p} = {00, 01, 10, 02, 20, 11, 12, 21, 22} with dim(M_{01}) = 7 = dim(M_{10}), dim(M_{02}) = 27 = dim(M_{20}), M_{11} ≅ M_{01} ⊗ M_{10}, M_{22} ≅ M_{02} ⊗ M_{20}, M_{12} ≅ M_{02} ⊗ M_{10} and M_{21} ≅ M_{01} ⊗ M_{20}. If γ denotes the nontrivial graph automorphism of G, then M_{10} ≅ M_{01}^{γ}, M_{20} ≅ M_{02}^{γ} and M_{12} ≅ M_{21}^{γ}. Using the 3-modular character table of G_{2}(3) listed in GAP [39], it follows that for q = 3 the tensor product β_{01} ⊗ β_{01} decomposes as

β_{01} ⊗ β_{01} = 1 · 1 + 2 · β_{01} + β_{10} + β_{02}. (4.2)

This is valid for arbitrary q = 3^{m}, since the weights λ ∈ X^{+} with λ ≤ 02 are exactly λ ∈ {00, 01, 10, 02}.

Because of dimensions, β_{01}^{σ} = β_{01}^{γ^{ν}α_{i}} for some ν ∈ {0, 1}, 0 ≤ i ≤ m − 1. Thus we get with Equation (4.2)

1 · 1 + 2 · β_{01}^{γ^{ν}α_{i}} + β_{10}^{γ^{ν}α_{i}} + β_{02}^{γ^{ν}α_{i}} = 1 · 1 + 2 · β_{01}^{σ} + β_{10}^{σ} + β_{02}^{σ}

and therefore

β_{10}^{γ^{ν}α_{i}} + β_{02}^{γ^{ν}α_{i}} = β_{10}^{σ} + β_{02}^{σ}.

Since the Brauer characters are linearly independent, it follows because of different dimensions that β_{10}^{σ} = β_{10}^{γ^{ν}α_{i}} and β_{02}^{σ} = β_{02}^{γ^{ν}α_{i}}. Thus hypothesis (i) of Proposition 2.11 follows for the case p = 3.

Now let p ≠ 3. Then dim(M_{01}) = 7 − δ_{p,2} and dim(M_{10}) = 14. If M is a simple kG-module with 1 < dim(M) < 18 then there exists an automorphism α of G, such that either M ≅ M_{01}^{α} or M ≅ M_{10}^{α}. This follows from [31, Prop. 5.4.12], since the center of G_{2}(q) is trivial and thus every simple kG-module is also a simple projective kG-module. Therefore, hypothesis (i) follows for p ≠ 3.

For the weak order ≺ in hypothesis (ii) we take the natural order relation ≤.

Part (B).

We have to show that if α is the group automorphism from Part (A), then σα^{−1} operates trivially on the characters which are p-exceptional.

In CHEVIE, the ordinary generic character tables of G_{2}(q) are divided into five cases. In all five cases, only the characters χ_{8} and χ_{9} are pexceptional.

Note that |G| = q^{6}(q − 1)^{2}(q + 1)^{2}(q^{2} − q + 1)(q^{2} + q + 1), and that G has maximal tori H_{3} ≅ ℤ_{q^{2}+q+1} and H_{6} ≅ ℤ_{q^{2}−q+1}.

q ≡ 1(3):

Let l be a rational prime dividing q^{2}−q+1. Because of (q^{2}−q+1, q−1) = 1, (q^{2} − q + 1, q + 1) = 1 and (q^{2} − q + 1, q^{2} + q + 1) = 1, a Sylow l-subgroup of H_{6} is also a Sylow l-subgroup of G, i.e. all Sylow l-subgroups of G are cyclic. Since χ_{8} and χ_{9} belong to the principal l-block and are not l-exceptional, σα^{−1} fixes χ_{8} and χ_{9} by Lemma 2.14.

q ≡ −1(3) or q ≡ 0(3):

Let l be a rational prime dividing q^{2} + q + 1. Similarly to q ≡ 1(3) it follows that all Sylow l-subgroups of G are cyclic. χ_{8} and χ_{9} belong to the principal l-block and are not l-exceptional. Thus by Lemma 2.14, σα^{−1} fixes χ_{8} and χ_{9}.

This proves Proposition 4.5. □

4.4. The finite twisted groups ^{3}D_{4}(q^{3}). The finite groups of type ^{3}D_{4} are the only missing simple exceptional groups of rank 2. ^{3}D_{4}(q^{3}) is simple for all q, and all group automorphisms are generated by inner and field automorphisms.

The generic character tables of ^{3}D_{4}(q^{3}) have been determined in [13] and can also be found in CHEVIE.

Proposition 4.6. The conjecture (ZC) is valid for ^{3}D_{4}(q^{3}) for all q = p^{m}, p a rational prime and m ≥ 1.

Proof. Let G = ^{3}D_{4}(q^{3}). Since there exist no ordinary irreducible characters of G which are p-exceptional, it suffices to look at:

Part (A).

The simple modules M_{λ_{i}} satisfy dim(M_{λ_{1}}) = 8 and dim(M_{λ_{2}}) = 28−2δ_{p,2}, M_{λ_{3}} ≅ M_{λ_{1}}^{α_{m}} and M_{λ_{4}} ≅ M_{λ_{1}}^{α_{2m}}. By [33, Thm. 1.1], for every simple kGmodule M with 1 < dim(M) < 32 there exists a group automorphism α of G such that either M ≅ M_{λ_{1}}^{α} or M ≅ M_{λ_{2}}^{α}. Note that [33, Thm. 1.1] shows this for the finite Chevalley groups of type D_{4}, but by Theorem 2.4, this must also be valid for G. This implies hypothesis (i) of Proposition 2.11. The weak order ≺ for hypothesis (ii) is defined as follows:

Let μ_{1} = b_{1}b_{2}b_{3}b_{4} and μ_{2} = c_{1}c_{2}c_{3}c_{4} be two different dominant weights. Then μ_{1} ≺ μ_{2} if and only if b_{1} + \frac{3}{2}b_{2} + b_{3} + b_{4} < c_{1} + \frac{3}{2}c_{2} + c_{3} + c_{4}, or b_{1} + \frac{3}{2}b_{2} + b_{3} + b_{4} = c_{1} + \frac{3}{2}c_{2} + c_{3} + c_{4} and b_{2} < c_{2}. Similarly to the case SL(3, q), it follows that μ_{1} ≤ μ_{2}, μ_{1} ≠ μ_{2} implies μ_{1} ≺ μ_{2} and μ_{2} ⊀ μ_{1}.

This proves Proposition 4.6. □

The Propositions 3.1, 4.2, 4.4, 4.5 and 4.6 together with the results of [4] prove Theorem 2.

Altogether the conjecture (ZC) is valid for the following finite groups of Lie type:

SL(2, p^{m}), PSL(2, p^{m}), p a rational prime
2B_{2}(2^{2m+1}), ^{2}G_{2}(3^{2m+1}), ^{2}F_{4}(2^{2m+1})
SL(3, p^{m}), PSL(3, p^{m}), p a rational prime
SU(3, p^{2m}), PSU(3, p^{2m}), p a rational prime
Sp(4, p^{m}), PSp(4, p^{m}) , p a rational prime
G_{2}(p^{m}), p a rational prime
3D_{4}(p^{3m}), p a rational prime.

Acknowledgments. The results of this paper except for the results concerning the series of type A_{2}, ^{2}A_{2} and B_{2} originate in my Ph.D. thesis [3]. I would like to express my thanks to Prof. Dr. W. Kimmerle and Prof. Dr. K.W. Roggenkamp for their continuous support.

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Received August 21, 1997 and revised February 9, 1998.

Southern Illinois University Carbondale, IL 62901-4408 E-mail address: fbleher@math.siu.edu

This paper is available via http://nyjm.albany.edu:8000/PacJ/1999/187-2-3.html.