First cohomology groups of Chevalley groups in cross characteristic.
type: Item
instance of: scholarly article
title: First cohomology groups of Chevalley groups in cross characteristic.
zbMATH Open document ID: 1237.20039
author: Robert Guralnick
author: Pham Huu Tiep
published in: Annals of Mathematics. Second Series
publication date: 2011-10-20T00:00:00Z
review text: Let \(G\) be a finite simple group of Lie type constructed over a field \(\mathbb F_q\) of characteristic \(p\). Let \(k\) be an algebraically closed field of characteristic \(r\) distinct from \(p\). (The cross characteristic case.) It is shown that very few irreducible \(kG\)-modules \(V\) have nonzero \(H^1(G,V)\). The authors also give an explicit upper bound for \(\dim H^1(G,V)\) that does not depend on \(q\) or \(V\), but only on the rank of the group. They obtain extremely strong bounds in the case that a Borel subgroup \(B\) has no fixed points in \(V\). They also bound the sum over all irreducible modules \(V\) of the product \(\dim V^B\cdot\dim H^1(G,V)\). Recall that Cline, Parshall and Scott showed that some bound on \(\dim H^1(G,V)\) exists in the equal characteristic case when \(r\mid q\).
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 5960710
zbMATH Keywords: cohomology of Chevalley groups
zbMATH Keywords: cross characteristic
zbMATH Keywords: cohomology groups
zbMATH Keywords: groups of Lie type
zbMATH Keywords: irreducible modules
MaRDI profile type: Publication
full work available at URL: annals.2011.174.1.16
OpenAlex ID: W2121765126
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DOI: 10.4007/ANNALS.2011.174.1.16
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description: scientific article; zbMATH DE number 5960710