On the cohomology of the finite Chevalley groups. (English) Zbl 0959.20017
Let \(G\) be a finite group and \(V\) a faithful irreducible \(G\)-module in characteristic \(p>0\). There is no loss of generality in assuming that \(V\) is absolutely irreducible. One of the basic problems in representation theory is to understand \(H^1(G,V)\). In particular, what is the dimension of \(H^1(G,V)\)? It has been shown that this question can be reduced to the case that \(G\) is a nonabelian simple group [see M. Aschbacher and L. Scott, J. Algebra 92, 44-80 (1985; Zbl 0549.20011) or M. Aschbacher and R. Guralnick, J. Algebra 90, 446-460 (1984; Zbl 0554.20017)].
The reviewer conjectured in 1984 that there was an absolute bound on \(\dim H^1(G,V)\). Aschbacher and the reviewer proved that \(\dim H^1(G,V)<\dim V\) [loc. cit.]. The reviewer improved this to \(\dim H^1(G,V)\leq(2/3)\dim V\) and the author and the reviewer improved the result to \(\dim H^1(G,V)\leq(1/2)\dim V\) [in Groups and geometries, Siena 1996, Trends in Mathematics 81-89 (1998; Zbl 0894.20039)]. The reviewed article was a key step in proving that result – better bounds are obtained for Chevalley groups (much better in the case of cross characteristic modules).
The main technique in the proof is to produce two subgroups \(S\) and \(T\) of \(G\) such that \(G=\langle S,T\rangle\), \(\dim H^1(S,V)\) and \(\dim H^1(T,V)\) are controlled and \(S\cap T\) is big and well understood in terms of the action on \(V\). This is particularly useful in the case of cross characteristic representations. This technique was used by J. L. Alperin and D. Gorenstein [Proc. Am. Math. Soc. 32, 87-88 (1972; Zbl 0249.18026)] to give a criterion for the vanishing of \(H^1\) under suitable hypotheses.
Different ideas are needed for modules in the natural representation and the bounds obtained are not as good. The factor of \(1/2\) is best possible in characteristics \(2\) and \(3\) (but only when \(\dim V=2\) in characteristic \(2\) and \(\dim V=4\) in characteristic \(3\)). The question still remains as to whether \(\dim H^1(G,V)\leq\ell\) for some absolute constant \(\ell\).
Chris McDowell and Leonard Scott in unpublished work have shown that for \(p\) sufficiently large, there exist modules \(V\) for \(G=\text{SL}(6,p)\) with \(\dim H^1(G,V)=3\). There are also examples of \(4\)-dimensional \(\text{Ext}^1(V,W)\) with \(V\) and \(W\) absolutely irreducible. Their methods involve explicit calculations of Kazhdan-Lusztig polynomials and use the facts that the Lusztig conjecture holds for \(p\) sufficiently large and that finite group cohomology and algebraic group cohomology are related. This brings into doubt the reviewer’s conjecture about an absolute bound on \(\dim H^1(G,V)\).
The reviewer conjectured in 1984 that there was an absolute bound on \(\dim H^1(G,V)\). Aschbacher and the reviewer proved that \(\dim H^1(G,V)<\dim V\) [loc. cit.]. The reviewer improved this to \(\dim H^1(G,V)\leq(2/3)\dim V\) and the author and the reviewer improved the result to \(\dim H^1(G,V)\leq(1/2)\dim V\) [in Groups and geometries, Siena 1996, Trends in Mathematics 81-89 (1998; Zbl 0894.20039)]. The reviewed article was a key step in proving that result – better bounds are obtained for Chevalley groups (much better in the case of cross characteristic modules).
The main technique in the proof is to produce two subgroups \(S\) and \(T\) of \(G\) such that \(G=\langle S,T\rangle\), \(\dim H^1(S,V)\) and \(\dim H^1(T,V)\) are controlled and \(S\cap T\) is big and well understood in terms of the action on \(V\). This is particularly useful in the case of cross characteristic representations. This technique was used by J. L. Alperin and D. Gorenstein [Proc. Am. Math. Soc. 32, 87-88 (1972; Zbl 0249.18026)] to give a criterion for the vanishing of \(H^1\) under suitable hypotheses.
Different ideas are needed for modules in the natural representation and the bounds obtained are not as good. The factor of \(1/2\) is best possible in characteristics \(2\) and \(3\) (but only when \(\dim V=2\) in characteristic \(2\) and \(\dim V=4\) in characteristic \(3\)). The question still remains as to whether \(\dim H^1(G,V)\leq\ell\) for some absolute constant \(\ell\).
Chris McDowell and Leonard Scott in unpublished work have shown that for \(p\) sufficiently large, there exist modules \(V\) for \(G=\text{SL}(6,p)\) with \(\dim H^1(G,V)=3\). There are also examples of \(4\)-dimensional \(\text{Ext}^1(V,W)\) with \(V\) and \(W\) absolutely irreducible. Their methods involve explicit calculations of Kazhdan-Lusztig polynomials and use the facts that the Lusztig conjecture holds for \(p\) sufficiently large and that finite group cohomology and algebraic group cohomology are related. This brings into doubt the reviewer’s conjecture about an absolute bound on \(\dim H^1(G,V)\).
Reviewer: Robert Guralnick (Los Angeles)
MSC:
| 20C33 | Representations of finite groups of Lie type |
| 20J06 | Cohomology of groups |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 20G10 | Cohomology theory for linear algebraic groups |
Keywords:
cohomology; Chevalley groups; finite groups of Lie type; irreducible modules; cross characteristic modulesReferences:
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