All-involution table algebras and finite projective spaces
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.
scientific article; zbMATH DE number 6489696
scientific article