On the range of non-vanishing \(p\)-torsion cohomology for \(\text{GL}_n(\mathbb{F}_p)\).
type: Item
instance of: scholarly article
title: On the range of non-vanishing \(p\)-torsion cohomology for \(\text{GL}_n(\mathbb{F}_p)\).
zbMATH Open document ID: 1063.20053
author: Adrian Barbu
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: For \(p\geq n\) the author constructs a nontrivial cohomology class in the cohomology group \(H^{2p-2}(\text{GL}_n(\mathbb{F}_p),\mathbb{F}_p)\). This is a remarkably low degree, so that these classes must be new. The method relies heavily on the Hecke algebra \({\mathcal H}(G//U)\) of \(G=\text{GL}_n(\mathbb{F}_p)\) with respect to its standard \(p\)-Sylow subgroup \(U\). This Hecke algebra is used to construct the classes over \(U\) and also to show that they extend to \(G\), by establishing that they are appropriate simultaneous eigenvectors for the action of \({\mathcal H}(G//U)\). Once the classes are constructed it is easy enough to show them nontrivial.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 2105338
zbMATH Keywords: Hecke algebras
zbMATH Keywords: punctual actions
zbMATH Keywords: cohomology classes
zbMATH Keywords: general linear groups
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.02.016
OpenAlex ID: W1980674898
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DOI: 10.1016/J.JALGEBRA.2004.02.016
description: scientific article; zbMATH DE number 2105338