Semisimplicity of adjacency algebras of association schemes
type: Item
instance of: scholarly article
title: Semisimplicity of adjacency algebras of association schemes
zbMATH Open document ID: 0942.05066
author: Akihide Hanaki
published in: Journal of Algebra
publication date: 2000-08-13T00:00:00Z
review text: Let \(Y=(X,\{R_i\}_{0\leq i\leq d})\) be an association scheme and \(n=|X|\). \(R\) be a commutative ring with 1. Define \(RY=\bigoplus_{i=0}^d Ra_i\) with the multiplication \(a_ia_j=\sum_k p_{ij}^ka_k\) and call \(RY\) the adjacency algebra of \(Y\) over \(R\). We call \(\{a_i\}\) the standard basis of \(Y\). We can define an \(RY\)-module \(RX\), which is an \(R\)-free module of rank \(n\) and indexed by \(X\). Let \(\mathbb{C}\) be the complex number field. Then \({\mathbb{C}}X=\bigoplus_{i=1}^r m_iS_i\), where \(S_i\) is an irreducible \({\mathbb{C}}Y\)-module. Put \(f_i= \dim S_i\). We define a frame number \({\mathcal F}(Y)\) by \[ {\mathcal F}(Y)=n^{d+1}\frac{\prod_{i=0}^d v_i}{\prod_{i=1}^r m_i^{f_i^2}}. \] The author investigates the semisimplicity of an adjacency algebra of association schemes over positive characteristic fields. Theorem 4.2. Let \(k\) be a field of characteristic \(p\). Then \(kY\) is semisimple if and only if the frame number \({\mathcal F}(Y)\) is not divided by \(p\).
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 1423606
zbMATH Keywords: association schemes
zbMATH Keywords: adjacency algebra
zbMATH Keywords: frame number
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: jabr.1999.8125
OpenAlex ID: W2092074718
cites work: On table algebras, c-algebras, and applications to finite group theory∗
cites work: Q3218140
cites work: The double cosets of a finite group
cites work: Q3035669
cites work: Q3826736
cites work: An algebraic approach to association schemes
DOI: 10.1006/JABR.1999.8125
description: scientific article; zbMATH DE number 1423606