Lagrange's theorem and integrality for finite commutative hypergroups with applications to strongly regular graphs
type: Item
instance of: scholarly article
title: Lagrange's theorem and integrality for finite commutative hypergroups with applications to strongly regular graphs
zbMATH Open document ID: 0855.43007
published in: Journal of Algebra
publication date: 1997-01-26T00:00:00Z
review text: The purpose of this paper is to show that algebraic concepts like integrality of coefficients may lead to new insights into the structure of finite hypergroups, and which consequences such results may have to related, more classical algebraic or combinatorial structures. The author defines a finite signed hypergroup \(K\) as an associative \(*\)-algebra over \(\mathbb{C}\) with basis \(K= \{c_0, c_1, \dots, c_n\}\) and properties similar to those of K. Ross or M. Rösler for ``topological'' hypergroups. Moreover, following R. I. Jewett, he requires for a finite hypergroup in addition that all structure constants are positive. It is now standard to define the Haar weights \(\omega (c_i)\) normalized by \(\omega (c_0) =1\) and to develop basic concepts of harmonic analysis for finite commutative signed hypergroups \(K\). This culminates in the fact that the dual \(\widehat {K}\) is again a finite commutative signed hypergroup with \(K^{\wedge \wedge} \simeq K\). A further section is devoted to subhypergroups \(D\) and quotient hypergroups \(K/D\) which can be formed in the finite case always in the natural way. These concepts lead to the following Lagrange-type result: If \(D\) is a subhypergroup of a finite commutative hypergroup such that all weights \(\omega (c_i)\) are integral for both \(K\) and \(\widehat {K}\), then one has \(\omega (D), \omega (K)\in \mathbb{N}\) with \(\omega (D) \mid \omega (K)\). Conjugacy class hypergroups \(K(G)\) of finite groups \(G\) motivate to say that a finite signed hypergroup is integral if the modified structure constants satisfy \[ {{\omega (c_i) \omega (c_j)} \over {\omega (c_k)}} n^k_{i,j}\in \mathbb{Z} \] for all \(i\), \(j\), \(k\). Unfortunately, character hypergroups \(K(\widehat {G})\) of finite groups \(G\) can fail to have this nice property. This observation leads the author to a modification of this integrality called resonance. It is shown that \(K(G)\) and \(K(\widehat {G})\) are resonant in this sense for all finite…
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 20N20
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 917722
zbMATH Keywords: Lagrange's theorem
zbMATH Keywords: integrality of coefficients
zbMATH Keywords: finite hypergroups
zbMATH Keywords: Haar weights
zbMATH Keywords: harmonic analysis
zbMATH Keywords: strongly regular graphs
zbMATH Keywords: Grassmannians
author: Norman John Wildberger
reviewed by: Michael Voit
MaRDI profile type: Publication
full work available at URL: jabr.1996.0158
OpenAlex ID: W1969500712
DOI: 10.1006/JABR.1996.0158
description: scientific article; zbMATH DE number 917722