The irreducible Brauer characters of the finite special linear groups in non-describing characteristics
type: Item
instance of: scholarly article
title: The irreducible Brauer characters of the finite special linear groups in non-describing characteristics
zbMATH Open document ID: 0948.20007
author: Jochen Gruber
published in: Journal of Algebra
publication date: 2000-08-31T00:00:00Z
review text: Let \(\widetilde G:=\text{GL}_n(q)\) and \(G:=\text{SL}_n(q)\). In the present paper the author studies the irreducible Brauer characters for \(G\) and primes \(\ell\) not dividing \(q\). They are described in terms of a basic set of ordinary characters of \(G\) which form a basis for the Abelian group of Brauer characters of \(G\). Their decomposition matrices are closely connected to the corresponding \(\ell\)-decomposition matrices for \(\widetilde G\) and the decomposition matrices of `extended \(q\)-Schur algebras' \({\mathcal S}({\mathcal H})\). These are defined in section two of the paper as endomorphism rings of certain modules of extended Hecke algebras of type \(A\). In particular, it is proved that these later decomposition matrices are lower unitriangular and even square lower unitriangular, if \({\mathcal S}({\mathcal H})\) is an extension of a \(q\)-Schur algebra which is coprime to \(\ell\). It is known that ordinary \(q\)-Schur algebras determine the decomposition matrix of \(\widetilde G\). The main theme of this article is to show how the extended \(q\)-Schur algebras \({\mathcal S}({\mathcal H})\) do the same for \(G\). Major tools in the \(\ell\)-modular representation theory of \(G\) and \(\widetilde G\) are the Gelfand-Graev characters, which are induced from non-degenerate characters of the unipotent (\(\ell'\)-)group \(U\) and therefore provide useful projective lattices. All needed results about those and their connections to cuspidal Brauer characters are collected in Section 3. As a major tool to link decomposition numbers with Harish-Chandra functors and series, the theory of `projective restriction systems' developed jointly with R. Dipper is introduced in Section 4. This theory can be formulated for groups with split BN-pair and thus it can be generalized to other finite groups of Lie type. It relies on a number of assumptions (e.g. purity of certain sublattices) which are later verified to hold for the useful submodules of the Gelfand…
reviewed by: Peter Fleischmann
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1295452
zbMATH Keywords: extended \(q\)-Schur algebras
zbMATH Keywords: irreducible Brauer characters
zbMATH Keywords: basic sets of ordinary characters
zbMATH Keywords: decomposition matrices
zbMATH Keywords: extended Hecke algebras
zbMATH Keywords: Gelfand-Graev characters
zbMATH Keywords: projective lattices
zbMATH Keywords: cuspidal Brauer characters
zbMATH Keywords: decomposition numbers
zbMATH Keywords: Harish-Chandra functors
zbMATH Keywords: groups with split BN-pairs
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: Lusztig series
MaRDI profile type: Publication
OpenAlex ID: W1975707113
cites work: Q3800275
cites work: On blocks of finite reductive groups and twisted induction
cites work: Q3907722
cites work: The characters of the group of rational points of a reductive group with non-connected centre.
cites work: Q3922829
cites work: Q3997911
cites work: On the Decomposition Numbers of the Finite General Linear Groups. II
cites work: Modular Harish-Chandra theory. I
cites work: Modular Harish-Chandra theory. II
cites work: The q ‐Schur Algebra
cites work: Representations of Hecke Algebras of General Linear Groups
cites work: On Lusztig's Isomorphism Theorems for Hecke Algebras and Representation Theory
cites work: The blocks of finite general linear and unitary groups
cites work: Basic Sets of Brauer Characters of Finite Groups of Lie Type II
cites work: Cuspidal unipotent Brauer characters
cites work: Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type
cites work: Q4851765
cites work: Decomposition numbers of finite classical groups for linear primes.
cites work: Harish‐Chandra Series of Brauer Characters in a Finite Group with a Split BN ‐Pair
cites work: Q4001632
cites work: Regular and Semisimple Blocks of Finite Reductive Groups
cites work: Representations of Generic Algebras and Finite Groups of Lie Type
cites work: Q4842690
cites work: The Decomposition Matrices of GL n (q ) for n ⩽ 10
cites work: On the values of characters of semi-simple groups over finite fields
cites work: Characters of Reductive Groups over a Finite Field. (AM-107)
cites work: Q3481947
DOI: 10.1006/JABR.1998.7736
description: scientific article; zbMATH DE number 1295452