A new family of modules with 2-dimensional 1-cohomology

John N. Bray, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS

                 E-mail: J.N.Bray@qmul.ac.uk.

27th July 2007

Abstract

It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, even faithful absolutely irreducible modules with 2-dimensional 1-cohomology are rare. We exhibit a new infinite family of such modules.

1 Introduction

Over the past few decades, a number of first cohomology groups have been computed, see for example [1, 2, 8, 5, 6]. The first cohomology group H^{1}(G; M) of a group G at a module M has two fundamental interpretations: first, it parametrises the complements of M in the split extension M:G of M by G; and second, it parametrises the extensions of M by the trivial G-module. In what follows we assume that M is a faithful absolutely irreducible module for G, since dim H^{1}(G; M) can be made arbitrarily large if either of these conditions fails.

In most cases which have been calculated up to now, H^{1}(G; M) has dimension 0 or 1, and in the few remaining cases it has dimension 2 or 3. The known examples of 3-dimensional 1-cohomology are given in Bray and Wilson [4] and Scott [9]. The two examples of [4] are explicit, whilst despite the infinitude of examples in [9], the results therein do not allow one to give an explicit example of a module with 3-dimensional 1-cohomology.

The results of this paper were discovered when I was carrying out a systematic calculation of 1-cohomologies (and 2-cohomologies) of (absolutely) irreducible modules of simple (and almost simple) groups, in connection with the Web-Atlas project [10]. Many of the 2-dimensional 1-cohomologies seen have an exceptional feel to them, such as dim H^{1}(A_{6}; 𝔽_{3}^{4}) = 2 (the smallest such example). During the course of this search I found that dim H^{1}(L_{3}(7); 𝔽_{3}^{55}) = 2, while for other irreducible L_{3}(7)-modules the 1-cohomology is at most 1-dimensional. This looked odd, especially when you compare the complexity of the Sylow 3-subgroup (3^{2}) with that of the Sylow 7-subgroup (7_{+}^{1+2}), and led to the Main Theorem. The smallest example which is part of this series, namely that dim H^{1}(L_{3}(4); 𝔽_{3}^{19}) = 2, was discovered earlier but somehow did not raise any eyebrows.

Main Theorem. Let n ⩾ 3, let p be a prime such that p | n, let q be a prime power such that q ≡ 1 (mod p), and let k be a field of characteristic p. Then dim_{k} H^{1}(L_{n}(q); M) = 2 where M is a certain absolutely irreducible kL_{n}(q)-module of dimension \frac{q^{n}−1}{q−1} − 2. We obtain M as the non-trivial composition factor of the permutation module of degree \frac{q^{n}−1}{q−1} for L_{n}(q) over k, corresponding to the action of SL_{n}(q) on the projective points (or hyperplanes) of the natural n-dimensional 𝔽_{q}SL_{n}(q)-module.

In addition to the above family of cross characteristic examples, the following examples of 2-dimensional 1-cohomology in defining characteristic are also known. Cline, Jones, Parshall and Scott [8, 5] give the examples of Ω_{4m}^{ε}(q) for q even, q > 2 and m ⩾ 2 acting on a module M of dimension 2m(4m − 1) − 2, where M is the non-trivial composition factor of Λ^{2}(V), with V being a natural module of Ω_{4m}^{ε}(q). They also point out that the ‘corresponding’ (i.e. 26-dimensional) module of ^{3}D_{4}(q) for q even, q > 2 also has 2dimensional 1-cohomology. The above results probably hold when q = 2. For the groups Ω_{4m}^{+}(2), m ⩾ 2 see [8, 5], while we have done explicit computations for the relevant modules of Ω_{4m}^{−}(2) for 4m ∈ {8, 12, 16, 20, 24, 28} and ^{3}D_{4}(2). In all cases, a subquotient of the tensor square of the natural module exhibits the 2-dimensional 1-cohomology.

Other than the above families of examples we know explicitly of only finitely many faithful absolutely irreducible modules for which the 1-cohomology is at least 2-dimensional.

2 Proof of the Main Theorem

Firstly, we establish some notation. The Atlas [7] notation is used for group structures, with E_{q} denoting an elementary abelian group of order q. For G a group and k a field, we also use k to denote the trivial kG-module. If V, U and W are modules, we write V ≅ U.W to mean that V has a submodule (isomorphic to) U and V/U ≅ W; and we write V ≅ U ·W if V ≅ U.W and U does not have a complement isomorphic to W. The standard results we quote here can be found, for example, in Benson’s book [3] (which states them in more generality).

Our example involves a certain module of L_{n}(q) in characteristic p, where n ⩾ 3, p | n and q ≡ 1 (mod p). We first recall some well-known facts about the groups L_{n}(q) = PSL_{n}(q) and SL_{n}(q). Firstly, for n ⩾ 2 and (n, q) ≠ (2, 2) or (2, 3) the groups L_{n}(q) are simple and the groups SL_{n}(q) are perfect and quasi-simple. Secondly, the Schur multiplier of SL_{n}(q) is always trivial, except for the cases (n, q) = (2, 4), (2, 9), (3, 2), (3, 4) or (4, 2), and the r′-part of the Schur multiplier of SL_{n}(q) is always trivial where r is the prime such that r | q.

It is known that the stabiliser of a projective point (i.e. 1-space) in SL_{n}(q) is a subgroup H of shape E_{q}^{n−1}:GL_{n−1}(q), which contains a subgroup E_{q}^{n−1}:SL_{n−1}(q) ≅ ASL_{n−1}(q) to index q − 1. We note that H need not be isomorphic to AGL_{n−1}(q); in particular, we have non-isomorphism in the cases of interest.^{1} Since n and n − 1 are coprime, the subgroup of scalars of SL_{n}(q) intersects the said subgroup ASL_{n−1}(q) trivially, and thus the image of this subgroup in any of the images C.L_{n}(q) of SL_{n}(q) is still isomorphic to ASL_{n−1}(q). Thus any non-trivial representation of SL_{n}(q) restricts faithfully to the subgroup ASL_{n−1}(q).

So now let n ⩾ 3, p | n, p | (q − 1), G ≅ SL_{n}(q), H ≅ E_{q}^{n−1}:GL_{n−1}(q) (the point stabiliser), and let k be a field of characteristic p. Let P be the permutation module over k of G on the cosets of H. Thus P is obtained by inducing the trivial kH-module up to G. Since Z(G) ⩽ H, Z(G) acts trivially on P, and thus P is a (permutation) module for L_{n}(q). The dimension of P is m := \frac{q^{n}−1}{q−1}. Permutation modules for transitive groups have a unique trivial submodule, U say, generated by the all 1s vector u = ∑_{i=1}^{m} e_{i}; and a unique trivial quotient, whose kernel is the augmentation submodule, W say, where W = { ∑_{i=1}^{m} a_{i}e_{i} : a_{i} ∈ k | ∑_{i=1}^{m} a_{i} = 0 }. We have U ⩽ W since p | m, and we let M := W/U. Now SL_{n}(q) is perfect and so there are no non-split modules of type k ·k for SL_{n}(q), and thus M has no trivial submodules or quotients. From Clifford Theory we find that a faithful representation of ASL_{n−1}(q) in characteristic p has dimension at least q^{n−1} − 1, since SL_{n−1}(q) has a single orbit on non-zero vectors of (the dual of) its natural module. But 2(q^{n−1} − 1) > dim M = \frac{q^{n}−1}{q−1} − 2 and so M is absolutely irreducible. Therefore P is a uniserial module of shape k ·M ·k.

Now SL_{n}(q) is perfect, and so dim H^{1}(SL_{n}(q); k) = 0, and dim H^{2}(SL_{n}(q); k) is the p-rank of the Schur multiplier of SL_{n}(q), which is 0. Now dim H^{1}(H; k) = 1, which is the prank of H/H′. The Eckmann–Shapiro Lemma then implies that dim H^{1}(SL_{n}(q); P) = 1. A well-known long exact sequence of cohomologies, applied to the permutation module P ≅ k ·M ·k, with (trivial) submodule U ≅ k and quotient Q ≅ M ·k is given below. The dimensions of these cohomology groups for SL_{n}(q) are given below. (The dimensions for L_{n}(q) differ from these, and also depend on whether the p-part of q − 1 is greater than the p-part of n.)

\begin{matrix}0 & → & H^{0}(k) & → & H^{0}(P) & → & H^{0}(Q) & → & H^{1}(k) & → & H^{1}(P) & → & H^{1}(Q) & → & H^{2}(k) & → & ⋯ \\ 0 & & 1 & & 1 & & 0 & & 0 & & 1 & & a(= 1) & & 0 \end{matrix}

1We note that the action of GL_{n−1}(q) on E_{q}^{n−1} is the [dual of] the tensor product of the natural representation and the determinant representation. As a result H ≅ E_{q}^{n−1}:GL_{n−1}(q) is isomorphic to AGL_{n−1}(q) if and only if H has trivial centre, which is if and only if (n, q − 1) = 1. In the cases of interest, we have (n, q − 1) ≠ 1, and thus H ≅ A̸GL_{n−1}(q).

We are interested in a := dim H^{1}(SL_{n}(q); Q), and the emboldened figures, which we justified earlier, ensure that a = 1. There is no module k ·k for SL_{n}(q) and thus dim H^{1}(SL_{n}(q); M) = 2. But any module M ·k for SL_{n}(q) actually represents the quotient group L_{n}(q), since k is not a submodule of M ⊗ k^{∗} ≅ M. (Alternatively, note that the centraliser algebra of the module M ·k consists just of scalar matrices. Thus central elements of SL_{n}(q) act as scalars on M ·k, and since they act trivially on the quotient k of this, they act trivially on M ·k.) Therefore we conclude that dim H^{1}(L_{n}(q); M) = 2, thus completing the proof of the Main Theorem.

The corresponding construction does not work when n = 2 (and thus p = 2). This is because the module M is not absolutely irreducible, splitting into two (absolutely irreducible) non-isomorphic summands of dimension \frac{1}{2}(q−1). This splitting always occurs over 𝔽_{4}, and will even occur over 𝔽_{2} if q ≡ ±1 (mod 8). The above argument gives dim H^{1}(L_{2}(q); M) = 2, and thus each constituent of M has 1-dimensional 1-cohomology.

Acknowledgements

We gratefully acknowledge the support of EPSRC grant GR/S41319. Robert Wilson has made suggestions that have improved the exposition of this paper.

References

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http://brauer.maths.qmul.ac.uk/Atlas/v3/.