Basic sets in defining characteristic for general linear groups of small rank.
type: Item
instance of: scholarly article
title: Basic sets in defining characteristic for general linear groups of small rank.
zbMATH Open document ID: 1173.20011
author: Olivier Brunat
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-10T00:00:00Z
review text: Let \(G\) be a finite group, let \(p\) be a prime and let \(\text{Irr}(G)\) be the set of ordinary irreducible characters of \(G\). For \(\chi\in\text{Irr}(G)\), let \(\widehat\chi\) denote the restriction of \(\chi\) to the \(p\)-regular elements of \(G\). A `\(p\)-basic set' of \(G\) is a subset \(B\) of \(\text{Irr}(G)\) such that the set \(\{\widehat\chi\mid\chi\in B\}\) is a \(\mathbb{Z}\)-basis of the \(\mathbb{Z}\)-module generated by \(\{\widehat\chi\mid\chi\in\text{Irr}(G)\}\). Examples show that finding a \(p\)-basic set can help with computing the decomposition matrix of a finite group \(G\). It is not known whether \(p\)-basic sets exist for all finite groups; cases that are known include finite groups of Lie type when \(p\) is different from the defining characteristic. This paper proposes a method for finding a \(p\)-basic set when \(G\) is a finite group of Lie type over a field of characteristic \(p\). Briefly, the method is to find certain ``good relations'' between characters of \(G\) (the author shows how such relations can be generated using Deligne-Lusztig theory and also the Gelfand-Graev character of \(G\)); these relations can then be used to find \(p\)-basic sets in some cases. The method is illustrated via explicit constructions of \(p\)-basic sets for \(G=\text{GL}_n(q)\), where \(q\) is a power of \(p\) and \(n=2\), \(3\) or \(4\).
reviewed by: Michael Bate
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5527491
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: irreducible characters
zbMATH Keywords: \(p\)-basic sets
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.09.006
OpenAlex ID: W2054214057
cites work: Q3682663
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DOI: 10.1016/J.JPAA.2008.09.006
description: scientific article; zbMATH DE number 5527491