Integral table algebras and Bose-Mesner algebras with a faithful nonreal element of degree three
type: Item
instance of: scholarly article
title: Integral table algebras and Bose-Mesner algebras with a faithful nonreal element of degree three
zbMATH Open document ID: 0958.05141
author: Harvey I. Blau
published in: Journal of Algebra
publication date: 2001-03-30T00:00:00Z
review text: Theorem 1. Let \((A,{\mathbf B})\) be a standard integral table algebra without nontrivial linear elements. If there exist a faithful nonreal element \(a\in {\mathbf B}\) of degree 3 and \(a\bar a=3\cdot {\mathbf 1}+b\) for some \(b\in \mathbf B\) with \(|b|=6\), then \({\mathbf B}\) is isomorphic \({\mathbf G}_n\) for some \(n\geq 3\) or \({\mathbf O}(S_3,{\mathbf Z}_m\times {\mathbf Z}_m)\) for some \(m\geq 4\). The combination of the main theorem of \textit{Z. Arad, E. Arisha, M. Fisman} and \textit{M. Muzychuk} [J. Algebra 231, No. 2, 473-483 (2000; review above)], the theorem B of \textit{H. I. Blau} (2000) and Theorem 1 yields Theorem 2. Let \((A,{\mathbf B})\) be a standard integral table algebra without nontrivial linear elements. If there exist a faithful nonreal element \(a\in {\mathbf B}\) of degree 3, then \({\mathbf B}\) is isomorphic to one of the following: a wreath product of \({\mathbf V}\) and \({\mathbf Z}_m\) for some \(m\geq 3\); \({\mathbf T}_n(3)\) for some \(n\geq 1\); \({\mathbf G}_n\) for some \(n\geq 3\); \({\mathbf O}({\mathbf Z}_3,H)\) for some abelian group \(H\) of rank at most 2; or \({\mathbf O}(S_3,{\mathbf Z}_m\times {\mathbf Z}_m)\) for some \(m\geq 3\).
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1526422
zbMATH Keywords: table algebra
zbMATH Keywords: Bose-Mesner algebra
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: jabr.1999.8188
OpenAlex ID: W2059728421
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DOI: 10.1006/JABR.1999.8188
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description: scientific article; zbMATH DE number 1526422