Operation of class sums on permutation modules
type: Item
instance of: scholarly article
title: Operation of class sums on permutation modules
zbMATH Open document ID: 0696.20012
DOI: 10.1016/S0747-7171(08)80005-3
author: T. W. Ostermann
author: Holger W. Gollan
published in: Journal of Symbolic Computation
publication date: 1990-01-01T00:00:00Z
review text: The authors present an efficient algorithm to compute the action of a conjugacy class sum of a finite group G on a permutation module V over a finite field F. This may be used to compute the block idempotents \(e_ i\)- knowing the classical character table and the distribution of characters into blocks - and the decomposition of V into blockwise components \(Ve_ i\). Thus, if one wants to know the irreducible constituents of V, the meataxe needs to be applied to smaller modules provided not all direct summands of V belong to the same block. As examples, the authors compute a) a 45-dimensional representation of \(PSL_ 3(4)\) over GF(9) starting with the permutation representation of degree 280 on the cosets of the Sylow 3-normalizer. b) a 27-dimensional representation of the Tits simple group \({}^ 2F_ 4(2)'\) over GF(5) starting with a permutation representation of degree 1755. The algorithm is based on the following key observation. Let \(V=F\Omega\) be a permutation module of G with \(\Omega =\{1,...,n\}\). For \(x\in G\) let K denote the conjugacy class of x in G and let \(\bar K\) denote the class sum of K. Then, the action of \(\bar K\) on V is given by the matrix \(A=(a_{ij})\) with \[ a_{ij}=| K| | \{(i,i^ x)| \quad i\in \Omega \}\cap \{(i^ g,j^ g)| \quad g\in G\}| /| \{i^ g,j^ g)| \quad g\in G\}| \] for \(i,j=1,...,n\). In case the action of G on \(\Omega\) is transitive, only the first row has to be computed.
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20H30
zbMATH DE Number: 4140019
zbMATH Keywords: computing direct summands of modules
zbMATH Keywords: efficient algorithm
zbMATH Keywords: conjugacy class sum
zbMATH Keywords: permutation module
zbMATH Keywords: block idempotents
zbMATH Keywords: irreducible constituents
zbMATH Keywords: permutation representation
zbMATH Keywords: Tits simple group
MaRDI profile type: Publication
cites work: Q3333220
cites work: Q3684278
cites work: The modular characters of the tits simple group and its automorphism group
cites work: Q3035669
cites work: Q3218280
description: scientific article; zbMATH DE number 4140019