Conjugacy class sums for induced modules: Construction and applications
type: Item
instance of: scholarly article
title: Conjugacy class sums for induced modules: Construction and applications
zbMATH Open document ID: 0813.20007
DOI: 10.1016/0024-3795(93)90240-O
author: Holger W. Gollan
published in: Linear Algebra and its Applications
publication date: 1994-03-13T00:00:00Z
review text: Let \(H\) be a subgroup of a finite group \(G\). The author shows how to compute a conjugacy class sum of an induced module \(M = \text{ind}^ G_ H(N)\), over some field \(F\), by using only one element of the conjugacy class. In the case where \(N\) is the trivial module of \(H\), \(M\) is the permutation module coming from the action of \(G\) on \(G/H\), and a formula for class sums of permutation modules has been derived in a previous paper by the author and \textit{T. W. Ostermann} [J. Symb. Comput. 9, No. 1, 39-47 (1990; Zbl 0696.20012)]. The idea for the case of a general \(H\)-module \(N\) is to realize the induced module \(M\) as the inverse image of a permutation module over some suitable set \(\Omega\) and use the class sum formula over this set \(\Omega\). This is first explained in the case where \(N\) is given by a one-dimensional representation \(\varphi: H \to F\). It is shown that one can take \(\Omega = G/H \times \varphi(H) \subset G/H \times F\). In the general case, a similar idea works to provide a set \(\Omega = G/H \times \varphi(H)\) where \(\varphi\) is an \(n\)-dimensional representation of \(H\) and \(\varphi(H) \subset \text{GL}_ n(F)\). Some comments are made on the computer implementation of this procedure for explicit computations. Further applications are given, to the problem of describing primitive linear groups of small degree by explicit matrices, and to the numerical computation of the rate of convergence of \(k\)-fold convolutions of probability measures to the uniform measure. This last application is illustrated by the example of the first Janko group \(J_ 1\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 484441
zbMATH Keywords: induced representation
zbMATH Keywords: finite group
zbMATH Keywords: conjugacy class sum
zbMATH Keywords: induced module
zbMATH Keywords: permutation modules
zbMATH Keywords: computer implementation
zbMATH Keywords: primitive linear groups
zbMATH Keywords: convolutions of probability measures
zbMATH Keywords: Janko group
reviewed by: Philippe Biane
MaRDI profile type: Publication
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full work available at URL: 0024-3795(93)90240-o
OpenAlex ID: W2062095979
description: scientific article; zbMATH DE number 484441