Basic sets of Brauer characters of finite groups of Lie type. III
type: Item
instance of: scholarly article
title: Basic sets of Brauer characters of finite groups of Lie type. III
zbMATH Open document ID: 0820.20018
DOI: 10.1007/BF02568193
author: Meinolf Geck
published in: Manuscripta Mathematica
publication date: 1995-01-30T00:00:00Z
full work available at URL: https://eudml.org/doc/156017
review text: This is the third in a series of papers investigating basic sets of Brauer characters of finite groups of Lie type [for part I see the author and \textit{G. Hiss}, J. Reine Angew. Math. 418, 173-188 (1991; Zbl 0771.20008) and for part II cf. J. Lond. Math. Soc., II. Ser. 47, 255-268 (1994; Zbl 0797.20013)]. In the first two papers the author asserted that ordinary basic sets always exist provided that \(\ell\) is good for \(G\), where \(G\) is a connected reductive group over \(\mathbb{F}_ q\); and that \(\ell\) does not divide the order of the group of components of the centre of \(G\). In this paper he studies the case where \(G\) is of classical type with connected centre and \(\ell = 2\), \(q\) is odd. Let \(G(\mathbb{F}_ q)\) be a finite classical group corresponding to \(G\). The author shows that there exists a set of irreducible characters of \(G(\mathbb{F}_ q)\) such that the corresponding matrix of scalar products with the characters of Kawanaka's generalized Gelfand-Graev representations is square unitriangular by using in an essential way Lusztig's theory of character sheaves. As an application he proves that there exists an ordinary basic set of 2- modular Brauer characters and that the decomposition matrix of the principal 2-block of \(G(\mathbb{F}_ q)\) has lower unitriangular shape.
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 714699
zbMATH Keywords: decomposition matrix of principal 2-block
zbMATH Keywords: basic sets of Brauer characters
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: connected reductive groups
zbMATH Keywords: finite classical groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: Gelfand-Graev representations
zbMATH Keywords: character sheaves
zbMATH Keywords: 2-modular Brauer characters
describes a project that uses: CHEVIE
MaRDI profile type: Publication
Wikidata QID: Q115392016
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description: scientific article; zbMATH DE number 714699