Tackling the generation problem in condensation.
type: Item
instance of: scholarly article
title: Tackling the generation problem in condensation.
zbMATH Open document ID: 1121.20009
author: Felix Noeske
published in: Journal of Algebra
publication date: 2007-04-11T00:00:00Z
review text: `Condensation' is an important computational tool in the modular representation theory of finite groups, which permits computation of properties of very large representations of groups by reducing the problem to computing with much smaller representations of suitable `condensation algebras'. The major problem with such methods, however, is the difficulty of finding a tractable set of elements in the algebra which can be proved to generate the algebra. In this paper, the author makes two important advances in this area. The first is a method using double cosets of the normaliser of the condensation subgroup to produce a relatively small set of elements which provably generate the algebra. The second is a method for testing whether a given set of elements does in fact generate the algebra. Both techniques can be used to prove that the small (algebra) representations have exactly the same structures as the large (group) representations, and thus have practical applications to determining (for example) degrees and Brauer characters of irreducible representations. The most spectacular applications are to sporadic simple groups.
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5141739
zbMATH Keywords: modular representations
zbMATH Keywords: condensation
zbMATH Keywords: generation problem
zbMATH Keywords: MeatAxe
zbMATH Keywords: computational representation theory
zbMATH Keywords: generating sets
zbMATH Keywords: double cosets
zbMATH Keywords: Brauer characters
zbMATH Keywords: irreducible representations
zbMATH Keywords: sporadic simple groups
reviewed by: Robert A. Wilson
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.06.021
OpenAlex ID: W1973905978
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DOI: 10.1016/J.JALGEBRA.2006.06.021
description: scientific article; zbMATH DE number 5141739