Scopes reduction and Morita equivalence classes of blocks in finite classical groups
type: Item
instance of: scholarly article
title: Scopes reduction and Morita equivalence classes of blocks in finite classical groups
zbMATH Open document ID: 0961.20036
author: Gerhard Hiss
author: Radha Kessar
published in: Journal of Algebra
publication date: 2001-02-21T00:00:00Z
review text: Donovan's conjecture says that, for a prime \(l\), the \(l\)-blocks of finite groups with a fixed defect fall into finitely many Morita equivalence classes. This conjecture refines an earlier conjecture by R.~Brauer. A stronger conjecture by Puig asserts that, for any finite \(l\)-group \(D\), there are only finitely many interior \(D\)-algebras (up to isomorphism) which are source algebras of \(l\)-blocks with defect group~\(D\). \textit{J.~Scopes} has verified Donovan's conjecture for blocks of symmetric groups [J. Algebra 142, No. 2, 441-455 (1991; Zbl 0736.20008)]. \textit{L.~Puig} then showed [Algebra Colloq. 1, No. 1, 25-55 (1994; Zbl 0830.20024)] that her argument also proves his own conjecture for blocks of symmetric groups. The method was extended to other classes of groups by various authors. In the paper under review, Puig's conjecture is shown to hold for unipotent \(l\)-blocks (where \(l\) is a unitary prime) of classical groups over a fixed finite field of characteristic \(p\neq l\), with some exceptions in the case of even-dimensional orthogonal groups. Moreover, an idea is outlined which (if successful) may lead to a verification of these conjectures for all finite simple groups.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1512973
zbMATH Keywords: unipotent blocks
zbMATH Keywords: defect groups
zbMATH Keywords: source algebras
zbMATH Keywords: Morita equivalences
zbMATH Keywords: Harish-Chandra induction
zbMATH Keywords: unipotent characters
zbMATH Keywords: Lusztig series
zbMATH Keywords: interior algebras
describes a project that uses: GAP
describes a project that uses: CHEVIE
MaRDI profile type: Publication
OpenAlex ID: W4213042261
cites work: Q3895655
cites work: On finite groups and their characters
cites work: Isométries de caractères et équivalences de Morita ou dérivées. (Isometries of characters and Morita or derived equivalences)
cites work: Generic Sylow theorems for the reductive groups over finite fields
cites work: Q4279639
cites work: On unipotent blocks and their ordinary characters
cites work: Q3682663
cites work: Q3997911
cites work: The q ‐Schur Algebra
cites work: The blocks of finite general linear and unitary groups
cites work: Q3803330
cites work: Brauer trees in classical groups
cites work: Q3971682
cites work: CHEVIE -- A system for computing and processing generic character tables
cites work: Decomposition numbers of finite classical groups for linear primes.
cites work: Representations of Generic Algebras and Finite Groups of Lie Type
cites work: Q4103728
cites work: Morita equivalence for blocks of finite general linear groups
cites work: Blocks and source algebras for the double covers of the symmetric and alternating groups
cites work: Source algebra equivalences for blocks of finite general linear groups over a fixed field
cites work: Equivalences for blocks of the Weyl groups
cites work: Donovan's conjecture, crossed products and algebraic group actions
cites work: A Counterexample to a Conjecture on the Cartan Invariants of a Group Algebra
cites work: Q3035669
cites work: The isomorphism problem for cyclic blocks and their source algebras
cites work: Irreducible representations of finite classical groups
cites work: Characters of Reductive Groups over a Finite Field. (AM-107)
cites work: Decomposition numbers of \(\text{Sp}(4,q)\)
cites work: Remarks on symbols, hooks and degrees of unipotent characters
cites work: Q5831368
cites work: Q4851773
cites work: On the local structure of Morita and Rickard equivalences between Brauer blocks
cites work: Cartan matrices and Morita equivalence for blocks of the symmetric groups
cites work: Q4846455
cites work: On the conjugacy classes in the unitary, symplectic and orthogonal groups
cites work: Sylow p-Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to p
DOI: 10.1006/JABR.2000.8319
description: scientific article; zbMATH DE number 1512973