On the vanishing ranges for the cohomology of finite groups of Lie type

2009/05/29 by Christopher P. Bendel, Bendel, Christopher P., Daniel K. Nakano +3
Mathematics · #20G10 #20J06 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20G10 #msc:20J06

paper · pdf · doi:10.48550/arxiv.0906.0026

arxiv created 2010/06/15 · arxiv updated 2010/06/16

Abstract

Let G(\mathbb F_q) be a finite Chevalley group defined over the field of q=p^r elements, and k be an algebraically closed field of characteristic p>0. A fundamental open and elusive problem has been the computation of the cohomology ring \opH^\bullet(G(\mathbb F_q),k). In this paper we determine initial vanishing ranges which improves upon known results. For root systems of type A_n and C_n, the first non-trivial cohomology classes are determined when p is larger than the Coxeter number (larger than twice the Coxeter number for type A_n with n>1 and r >1). In the process we make use of techniques involving line bundle cohomology for the flag variety G/B and its relation to combinatorial data from Kostant Partition Functions.

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