Dade’s inductive conjecture for the Ree groups of type G_{2} in the defining

characteristic Charles W. Eaton ^{1} Department of Mathematics and Computer Science University of Leicester Leicester LE1 7RH England email: cwe1@mcs.le.ac.uk

Abstract

We verify the inductive form of Dade’s conjecture for the finite simple groups 2G_{2}(3^{2m+1}), where m is a positive integer, for the prime p = 3. Together with [2] this completes the verification of the conjecture for this series of groups.

1: INTRODUCTION

In [7], [8] and [9] Dade presents a series of conjectures, the strongest of which (the inductive conjecture) he claims admits a reduction to finite simple groups. The conjectures originate from the reformulation of Alperin’s weight conjecture (see [1]) given by Knörr and Robinson in [11], and consequently a verification of any of Dade’s conjectures implies that Alperin’s weight conjecture holds. It should be noted however that the verification of Dade’s conjecture for a given group does not imply that Alperin’s conjecture holds for that group. However, Alperin’s conjecture is known to hold for finite groups with split BN-pairs in the defining characteristic and so does indeed hold in this case (see [4]).

Let G be a finite group and p a prime. We say that a p-subgroup Q of G is radical if Q = O_{p}(N_{G}(Q)), where O_{p}(H) is the unique maximal normal p-subgroup of H. We say that a chain σ : Q_{0} < ⋯ < Q_{n} of p-subgroups of G (where inclusions are strict) has length |σ| = n, and write G_{σ} = N_{G}(Q_{0}) ∩ ⋯ ∩ N_{G}(Q_{n}) for its stabilizer under the conjugation action of G. Write σ_{i} for the p-chain Q_{0} < ⋯ < Q_{i} and V_{σ} = Q_{0}. We say that the p-chain σ is radical if Q_{i} = O_{p}(N_{G}(σ_{i})) for each i, i.e., if Q_{0} is a radical p-subgroup of G and Q_{i} is a radical p-subgroup of N_{G}(σ_{i−1}) for each i ≠ 0. Write R = R(G) for the set of radical p-chains of G and write R(G|Q) = {σ ∈ R(G) : V_{σ} = Q}.

1Partially supported by EPSRC award no. 95700575

Write R(G|Q)/G for a set of orbit representatives under the action of G. If N ◁ G and μ ∈ Irr(N) (the set of irreducible characters of N), then denote by Irr(G, μ) the set of irreducible characters of G covering μ.

Let R be a local principal ideal domain whose residue field k = R/J(R) has characteristic p and whose field of fractions K has characteristic zero and is a splitting field for G. For example, we could take (K, R, k) to be a complete p-modular system in which R contains a primitive |G|-th root of unity.

As mentioned in [9], it suffices in verifying Dade’s inductive conjecture for the groups^{2}G_{2}(3^{2m+1}) to test them only for the invariant conjecture, since Out(^{2}G_{2}(3^{2m+1})) is cyclic and ^{2}G_{2}(3^{2m+1}) has trivial Schur multiplier (see [6]). We state here only the invariant conjecture, and refer the reader to [9] for a statement of the inductive conjecture.

Conjecture 1 (Dade’s invariant) Let B be a p-block of positive defect of a finite group G satisfying O_{p}(G) = 1, and suppose that G ◁ E. Write Irr_{d}(G_{σ}, B) for the set of irreducible characters χ of G_{σ} lying in p-blocks which are Brauer correspondents of B and whose degrees satisfy χ(1)_{p}p^{d} = |G|_{p} (we say that χ has defect d). If G◁H ≤ E and σ ∈ R(G), then denote by Irr_{d}(G_{σ}, B, H) the set of those characters in Irr_{d}(G_{σ}, B) with inertial subgroup N_{H}(σ) in N_{E}(σ). Write k_{d}(G_{σ}, B, H) = |Irr_{d}(G_{σ}, B, H)|. Then

∑_{σ∈R(G|1)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, H) = 0.

Jianbei An has verified the inductive conjecture for the groups ^{2}G_{2}(3^{2m+1}) for all primes except p = 3 (see [2]), and so the results presented here complete the verification of the conjecture for these groups.

Henceforth let G = ^{2}G_{2}(3^{2m+1}). We make the trivial observation that G has a split BN-pair of rank one and so the only radical 3-subgroups are the trivial group 1 and the Sylow 3-subgroups of G. Let P ∈ Syl_{3}(G) be the unipotent radical, so that R(G|1)/G may be taken to be {1, 1 < P } and the conjecture reduces to checking that k_{d}(G, B, H) = k_{d}(N_{G}(P), B, H) for each choice of B, H and d.

In Section 2 we investigate the irreducible characters of N_{G}(P), the characters of G having been investigated fully in [15]. In Section 3 we examine the action of the automorphisms of G on the irreducible characters of G and N_{G}(P), and so verify the conjecture in this case. Blau and Michler have verified a simpler form of the conjecture for these groups, and given the action of outer automorphisms on the 3-regular classes (see [3]). Our results may be viewed as an extension of theirs.

2: IRREDUCIBLE CHARACTERS OF N_{G}(P)

We realise G as a twisted Chevalley group (see for example [13] or [5]). Write q = 3^{2m+1}, where m > 0.

Lemma 2 |P | = q^{3} and we may label its elements as x(t, u, v), where 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}^{3f}, v_{1} + v_{2} − t_{2}u_{1} + t_{1}t_{2}^{3f+1} − t_{1}^{2}t_{2}^{3f}),

where f = 3^{m}.

Conjugation within P 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_{1}t_{2}^{3f} + t_{2}t_{1}^{3f}, v_{2} − t_{2}u_{1} + t_{1}u_{2} + t_{1}t_{2}^{3f+1} + t_{1}^{2}t_{2}^{3f} + t_{1}^{3f}t_{2}^{2}).

Z(P) = {x(0, 0, v) | v ∈ 𝔽_{q}} and |Z(P)| = q. The subgroup

P_{1} = {x(0, u, v) ∈ P | u, v ∈ 𝔽_{q}}

of order q^{2} is normal in P and P_{1} = P′ = Φ(P), so P/P_{1} is elementary abelian.

Proof The description of the elements and their multiplication follows from [5]. From this it is easily seen that x(t, u, v)^{−1} = x(−t, −u − t^{3f+1}, −v − tu + t^{3f+2}) for each t, u, v ∈ 𝔽_{q}, and that the conjugation action is as given. The rest follows easily.

We may calculate the conjugacy class lengths explicitly:

Lemma 3 P has q conjugacy classes of length 1, q − 1 of length q and 3(q − 1) of length \frac{1}{3}q^{2}.

Proof Note that

x(t_{1}, u_{1}, v_{1})x(0, u_{2}, 0)x(t_{1}, u_{1}, v_{1})^{−1} = x(0, u_{2}, t_{1}u_{2}),

so that each x(0, u_{2}, 0) with u_{2} ∈ 𝔽_{q}^{×} lies in a distinct conjugacy class of length q contained in P_{1}. It is clear that we have q conjugacy classes of length one, thus accounting for every element of P_{1}.

Suppose now that x(t_{2}, u_{2}, v_{2}) ∈ P with t_{2} ≠ 0. If x(t_{1}, u_{1}, v_{1}) ∈ C_{P} (x(t_{2}, u_{2}, v_{2})), then t_{2}t_{1}^{3f} = t_{1}t_{2}^{3f}. Let ϕ : 𝔽_{q} → 𝔽_{q} be the additive endomorphism sending λ to t_{2}λ^{3f} − λt_{2}^{3f}. If 0 ≠ λ ∈ ker(ϕ), then λ^{3f} = λt_{2}^{3f−1}. Without loss of generality, assume t_{2}^{3f−1} = 1, so λ^{3f} = λ. Hence

λ^{3^{m+1}−1} = 1 = λ^{q−1} = λ^{3^{m}(3^{m+1}−1)+3^{m}−1}

and

1 = λ^{q−1} = λ^{3(3^{m}−1)(3^{m}+1)+2} = λ^{2},

so |ker(ϕ)| = 3. This gives |C_{P} (x(t_{2}, u_{2}, v_{2}))| = 3q since we can choose u_{1} to give t_{2}u_{1} any value in 𝔽_{q}. Hence we have found 3(q − 1) conjugacy classes of length \frac{1}{3}q^{2}. This accounts for all the elements of P.

We use the action of P on Irr(P_{1}) to calculate the nature and degrees of the irreducible characters of P.

Lemma 4 P has q linear characters, q−1 irreducible characters of degree q and 3(q−1) of degree 3^{m}.

Proof Considering 𝔽_{q} as an n = (2m + 1)-dimensional vector space over 𝔽_{3}, write ζ ∈ 𝔽_{q} as ζ = (ζ_{1}, …, ζ_{n}). Write the elements of Irr(P_{1}) as χ_{x,y}, x, y ∈ 𝔽_{q}, where, for x(0, u, v) ∈ P_{1},

χ_{x,y} (x(0, u, v)) = ∏_{k=1}^{n} e^{\frac{2u_{k}x_{k}πi}{3}} ∏_{j=1}^{n} e^{\frac{2v_{j} y_{j} πi}{3}} .

Recall that

x(t_{1}, u_{1}, v_{1})x(0, u_{2}, 0)x(t_{1}, u_{1}, v_{1})^{−1} = x(0, u_{2}, t_{1}u_{2}) ∀ t_{1}, u_{1}, v_{1}, u_{2} ∈ 𝔽_{q}.

From this we see that for any x ∈ 𝔽_{q}, we have I_{P} (χ_{x,0}) = P and that for any x ∈ 𝔽_{q}, y ∈ 𝔽_{q}^{×}, we have I_{P} (χ_{x,y}) = P_{1}. This gives us q orbits of length one and q − 1 orbits of length q. Representatives of the long orbits are χ_{0,y}, y ∈ 𝔽_{q}^{×}.

By Clifford theory representatives of the long orbits induce to give distinct irreducible characters of P of degree q. Also χ_{0,0} = 1_{P_{1}} extends to P in q distinct ways to give q linear characters of P. Since P_{1} is the derived subgroup of P, every linear character of P is an extension of χ_{0,0}.

Now consider the characters χ_{x,0}, where x ∈ 𝔽_{q}^{×}. We show that all irreducible characters of P covering such characters of P_{1} have the same degree.

Fix μ = χ_{x,0}, x ∈ 𝔽_{q}^{×}. Since I_{P} (χ_{x,0}) = P, there is a degree preserving 1-1 correspondence Irr(P, μ) ↔ Irr({ \atop (P/P_{1})}, μ̂), where { \atop (P/P_{1})} is a central extension of P/P_{1} with

kernel A (i.e., A ≤ Z({ \atop (P/P_{1})}) ∩ { \atop (P/P_{1})}′ and { \atop (P/P_{1})}/A ≅ P/P_{1}) and μ̂ ∈ Irr(A) (see,

for example, Chapter 11 of [10]). We may take A to be a cyclic p-group (see [12]). However, the Schur multiplier of an elementary abelian group is itself elementary abelian, and so |A| = 3. Note that μ̂ is non-trivial since we may not extend μ to P (as all the linear characters of P cover χ_{0,0}). This demonstrates that the degrees of the elements of Irr(P, χ_{x,0}) are independent of the choice of x ∈ 𝔽_{q}^{×}. Since P possesses 5q − 4 conjugacy classes we then have |Irr(P, χ_{x,0})| = 3, and so all irreducible characters of P lying over χ_{x,0} must have the same degree, which must be 3^{m}, as required.

We examine the action of N_{G}(P) on the irreducible characters of P to calculate the irreducible character degrees of N_{G}(P). We note that by using Ward’s character table for G (see [15]) and the fact that the Knörr-Robinson reformulation of Alperin’s weight conjecture is known to hold for finite groups with split BN-pairs in the defining characteristic (see [11, 5.3]), we may already observe that N_{G}(P) possesses q + 7 conjugacy classes.

Lemma 5 N_{G}(P) has irreducible characters of degrees 1, q − 1, 3^{m}(q − 1)/2, 3^{m}(q − 1), q(q − 1) with multiplicity q − 1, 1, 4, 2, 1 respectively.

Proof By the results of [13] N_{G}(P) = P W, where W ≅ 𝔽_{q}^{×} acts transitively on the nontrivial elements of P/P_{1} and of Z(P). Hence W acts transitively on the q−1 non-trivial linear characters of P, and so by Clifford’s theorem we obtain q − 1 linear characters of N_{G}(P) and one irreducible character of degree q − 1 (where the linear characters are the irreducible characters of N_{G}(P)/P regarded as characters for N_{G}(P), and the irreducible character of degree q − 1 is λ^{N_{G}(P)}, where λ is a non-trivial linear character of P). The irreducible characters of degree q of P are also permuted transitively by W, giving one irreducible character of degree q(q − 1). Since k(N_{G}(P)) = q + 7 we are left with 6. The remaining irreducible characters must cover the irreducible characters of P of degree 3^{m}.

By Ree [13], W possesses an unique involution, h_{0}, and C_{P} (h_{0}) = {x(0, u, 0)|u ∈ 𝔽_{q}}. Hence W/ < h_{0} > acts on P_{1}/Z(P) with 2 orbits of length (q − 1)/2 and one of length one (noting that by [13] W stabilizes P_{1}). It follows from Brauer’s theroem that W acts on the irreducible characters χ_{x,0}, where x ∈ 𝔽_{q}^{×}, with two orbits, which must then both be of length (q − 1)/2. Recall that each of these χ_{x,0} is covered by three irreducible characters of P. It follows that the irreducible characters of N_{G}(P) of height m must consist of two of degree 3^{m}(q − 1) and four of degree 3^{m}(q − 1)/2.

3: DADE’S CONJECTURE

Blau and Michler give a description of the action of an arbitrary outer automorphism on the semisimple classes of G:

Lemma 6 (Blau and Michler [3]) Out(G) is cyclic and consists of automorphisms of the form τ^{r}, where τ is the Frobenius automorphism x → x^{3}. We have |Out(G)| = 2m + 1, and τ^{r} stabilizes P.

When r is an integer dividing 2m + 1, τ^{r} fixes 3^{r} 3-regular conjugacy classes of G.

We give a combinatorial lemma included in [3]:

Lemma 7 ( [3]) Let A be a finite group acting on two sets S_{1} and S_{2} as a permutation group. Let H ≤ A and write f_{i}(H) = |{s ∈ S_{i}|s^{a} = s ∀ a ∈ H}| and m_{i}(H) = |{s ∈ S_{i}|C_{A}(s) = H}|.

If f_{1}(H) = f_{2}(H) for all H ≤ A then m_{1}(H) = m_{2}(H) for all H ≤ A.

We come to the main result of the paper:

Theorem 8 Dade’s inductive conjecture holds for G for the prime p = 3.

Proof As remarked earlier it suffices to verify Conjecture 1 for G. By [15] G possesses only one 3-block of defect zero, namely that containing the Steinberg character, and C_{G}(P) ⊂ P. Hence there is only one 3-block of maximal defect (by a well-known result of block theory). But defect groups are radical p-subgroups and the Sylow 3-subgroups are the only non-trivial radical 3-subgroups of G, so the principal 3-block B is the unique 3-block of non-zero defect. Suppose that G ◁ E and choose t|2m + 1 such that τ^{t} generates E/G, where τ is as in Lemma 6. Let G ◁ H ≤ E, and choose r|2m + 1 such that τ^{r} generates H/G.

From the character table for G (see [15, p.87]), we have k_{6m+3}(G, B) = q, k_{5m+3}(G, B) = 6 and k_{4m+2}(G, B) = 1, this accounting for all the irreducible characters of G in B.

Consider first the action of τ^{r} on Irr(G, B). Examination of the character table given in [15] gives us that τ^{r} fixes every 3-singular conjugacy class of G, hence by Lemma 6 τ^{r} fixes 3^{r}+8 conjugacy classes of G. So τ^{r} fixes 3^{r}+7 irreducible characters of B, by Brauer’s theorem. Further examination of the character table and using the fact that τ^{r} has odd order reveals that τ^{r} fixes every irreducible character of B of positive height, and so 3^{r} irreducible characters of height zero, i.e., f_{Irr_{6m+3}(G,B)}(⟨τ^{r}⟩) = 3^{r}, f_{Irr_{5m+3}(G,B)}(⟨τ^{r}⟩) = 6 and f_{Irr_{4m+2}(G,B)}(⟨τ^{r}⟩) = 1 in the notation of Lemma 7.

Now consider the action of τ^{r} on Irr(N_{G}(P)). The 3-regular conjugacy classes of N_{G}(P) = P W are represented by the elements of W, and so τ^{r} fixes 3^{r} − 1 3-regular conjugacy classes of N_{G}(P). Hence τ^{r} fixes 3^{r} − 1 irreducible Brauer characters of N_{G}(P), and so 3^{r} − 1 linear characters and the irreducible character of degree q − 1. It remains to examine the action on the characters of positive height. Clearly the unique irreducible character of degree q(q −1) is fixed by τ^{r}, as are the two of degree 3^{m}(q −1) since τ^{r} has odd order. Finally it is clear from the construction of the remaining characters that τ^{r} cannot act on the four irreducible characters of degree 3^{m}(q − 1)/2 with an orbit of length three, and so τ^{r} fixes these also. Hence f_{Irr_{6m+3}(N_{G}(P),B_{0})}(⟨τ^{r}⟩) = 3^{r}, f_{Irr_{5m+3}(N_{G}(P),B_{0})}(⟨τ^{r}⟩) = 6 and f_{Irr_{4m+2}(N_{G}(P),B_{0})}(⟨τ^{r}⟩) = 1.

The result then follows from Lemma 7.

Remark The above results also yield a verification of Robinson’s conjecture (see [14]) for G.

ACKNOWLEDGEMENTS

I would like to thank my Ph.D. supervisor Geoffrey Robinson for his sound guidance. I would also like to thank R. Baddeley for his advice concerning Chevalley groups.

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