Morita equivalence for blocks of finite general linear groups
type: Item
instance of: scholarly article
title: Morita equivalence for blocks of finite general linear groups
zbMATH Open document ID: 0896.20009
DOI: 10.1007/BF02567944
author: Thomas Jost
published in: Manuscripta Mathematica
publication date: 1998-10-06T00:00:00Z
full work available at URL: https://eudml.org/doc/156219
review text: In modular representation theory of finite groups, one of the prominent problems is to find invariants of a block of a group algebra which serve as a measure for the complexity of its module category. Consequently, one may inquire how well these invariants control the structure of the module category of a block. One such invariant is the defect group of a block. P. Donovan [\textit{J. P. Alperin}, Proc. Symp. Pure Math. 37, 369-375 (1980; Zbl 0449.20019)]\ conjectured that up to Morita equivalence there is only a finite number of block algebras with defect group isomorphic to a given \(r\)-group. The conjecture was answered affirmatively by J. Scopes in 1990 for blocks of the symmetric groups by using combinatorial techniques of the representation theory of these groups. In the paper under review, by generalizing Scopes' method, the author shows that for a given weight vector \(w\) and a given \(q\) there is only a finite number of Morita equivalence classes of blocks with weight vector \(w\), for \(R\text{GL}(n,q)\) in the nondescribing characteristic. He considers \(R\in\{{\mathfrak R},F\}\), where \({\mathfrak R}\) is a complete discrete valuation ring whose residue class field \(F\) is an algebraically closed field whose characteristic \(r\) is coprime to \(q\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 980703
zbMATH Keywords: complexity of module categories
zbMATH Keywords: invariants
zbMATH Keywords: Morita equivalence
zbMATH Keywords: block algebras
zbMATH Keywords: defect groups
zbMATH Keywords: Morita equivalence classes of blocks
zbMATH Keywords: weight vectors
reviewed by: Qaiser Mushtaq
MaRDI profile type: Publication
cites work: Q3682663
cites work: Q4724810
cites work: Modular Harish-Chandra theory. I
cites work: Identification of the irreducible modular representations of \(GL_n(q)\).
cites work: The q ‐Schur Algebra
cites work: The blocks of finite general linear and unitary groups
cites work: Q3971682
cites work: Schur indices of some groups of Lie type
cites work: The Characters of the Finite General Linear Groups
cites work: Q4103728
cites work: The representation theory of the symmetric groups
cites work: The Irreducible Representations of the Finite General Linear Groups
cites work: Q3952291
cites work: Q3792865
cites work: On the Schur indices of GL(n,q) and SL(2n+1,q)
cites work: Cartan matrices and Morita equivalence for blocks of the symmetric groups
cites work: Sylow p-Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to p
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description: scientific article; zbMATH DE number 980703