Primitivity of permutation groups, coherent algebras and matrices
type: Item
instance of: scholarly article
title: Primitivity of permutation groups, coherent algebras and matrices
zbMATH Open document ID: 1011.20003
author: Gareth Jones
author: Yossi Moshe
author: Mikhail H. Klin
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-02-27T00:00:00Z
review text: The term ``primitive'' is used in many senses in algebra. In particular, a permutation group is primitive if it preserves no non-trivial equivalence relation; a coherent algebra (such as the centralizer algebra of a permutation group) is primitive if it does not contain the matrix of such a relation; and a real square matrix is primitive if some power of the matrix has all entries strictly positive. There is a natural relationship between the first two uses of the term primitive. The main result of the present paper is the following relationship between the last two uses. A coherent algebra \(W\) has a basis consisting of square \(\{0,1\}\)-matrices. Suppose that \(W\) is not the centralizer algebra of a regular permutation group of prime degree. Then \(W\) is a primitive coherent algebra if and only if each nonidentity basis element is a primitive matrix. This result is applied to give necessary and sufficient conditions for the exponentiation \(W\uparrow G\) of a coherent algebra \(W\) by a permutation group \(G\) to be primitive.
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1760678
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: centralizer algebras
zbMATH Keywords: primitive coherent algebras
zbMATH Keywords: primitive matrices
zbMATH Keywords: exponentiations
zbMATH Keywords: wreath products
reviewed by: John D. Dixon
MaRDI profile type: Publication
full work available at URL: jcta.2001.3245
OpenAlex ID: W2045385512
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DOI: 10.1006/JCTA.2001.3245
description: scientific article; zbMATH DE number 1760678