All-involution table algebras and finite projective spaces
type: Item
instance of: scholarly article
title: All-involution table algebras and finite projective spaces
zbMATH Open document ID: 1359.16014
author: Harvey I. Blau
author: Gang Chen
published in: Archiv der Mathematik
publication date: 2015-10-05T00:00:00Z
review text: Table algebras appear in diverse places: as group algebras, centers of group algebras, Hecke (double coset) algebras, and character rings for finite groups; as adjacency (Bose-Mesner) algebras of association schemes; and as Grothendieck rings for fusion categories. Note that an involution, as defined by \textit{P.-H. Zieschang} [Theory of association schemes. Berlin: Springer (2005; Zbl 1079.05099)] for adjacency algebras, is a nonidentity basis element \(b\) such that \(b^2\) is a linear combination of \(1\) and \(b\) (possibly just a scalar times \(1\) when \(b\) is linear -- also called thin or group-like). Standard integral table algebras (SITAs) that are generated by a set \(\mathbf{I}\) of involutions are called Coxeter algebras if they satisfy two properties, constraint and the exchange condition, defined by Zieschang [loc. cit.]. The goal of this article is to show that all-involution SITAs have their own classification. Let \({\mathbf B}\) be the distinguished basis of such a table algebra \(A\), and assume that \(|\mathbf{B}|>2\). If two involutions generate \(\mathbf{B}\), then \(\mathbf{B}\) is one of the algebras \({\mathbf H}_n\), defined for all integers \(n\geq 3\) in [the authors, Algebr. Represent. Theory 17, No. 5, 1527--1552 (2014; Zbl 1308.20007)]. If more than two involutions are required to generate \(\mathbf{B}\), then the elements of \(\mathbf{B}\{1\}\) correspond to the points of a finite projective space, whose incidence geometry determines the algebra multiplication.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 6489696
zbMATH Keywords: table algebra
zbMATH Keywords: projective space
zbMATH Keywords: involution
zbMATH Keywords: association scheme
reviewed by: Feng Wei
MaRDI profile type: Publication
full work available at URL: s00013-015-0795-9
OpenAlex ID: W1927386226
cites work: Generalized table algebras
cites work: Table algebras
cites work: Table bases as unions of proper closed subsets.
cites work: Sylow theory for table algebras, fusion rule algebras, and hypergroups.
cites work: A characterization of association schemes from affine spaces
cites work: Some implications on amorphic association schemes
cites work: The Conjugacy Class Number k(G): A Different Perspective
cites work: Q4284635
cites work: Q3998721
cites work: A note on conjugacy classes of finite groups.
cites work: Q5733090
cites work: An algebraic approach to association schemes
cites work: Theory of Association Schemes
DOI: 10.1007/S00013-015-0795-9
Recommended article: Q3356510
Recommended article: Q4387383
Recommended article: Table algebras with central fusions
Recommended article: A construction of integral standard generalized table algebras from parameters of projective geometries
Recommended article: Common properties of table algebras and association schemes
Recommended article: On a class of integral table algebras
Recommended article: Some structure theory of table algebras and applications to association schemes
Recommended article: On \(p\)-standard table algebras of order \(p^3\). II
Recommended article: Table algebras with exactly one irreducible character whose degree and multiplicity are not equal
Recommended article: Q4374879
description: scientific article; zbMATH DE number 6489696