Coherent algebras
type: Item
instance of: scholarly article
title: Coherent algebras
zbMATH Open document ID: 0618.05014
DOI: 10.1016/S0024-3795(87)90326-0
author: D. G. Higman
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: For a finite nonempty set X, a coherent algebra on X is defined to be a subalgebra of the algebra \(M_ X(C)\), the X-square matrices over the complex numbers \({\mathbb{C}}\), which is closed under the adjoint map and under Hadamard multiplication and contains the matrix with 1 in each entry. (Here Hadamard multiplication means \((a_{ij})(b_{ij})=(a_{ij}b_{ij}).)\) A configuration \(C=(X,(f_ i)_{i in I})\) on X over a finite set I consists of the set X together with a family of nonempty binary relations on X and as such can be identified with the family \((A_ i)_{i in I}\) of matrices of the \(f_ i\), called the adjacency matrices of C. This paper studies so- called coherent configurations (see the author [Geom. Dedicata 4, 1-32 (1975; Zbl 0333.05010) and 5, 413-424 (1976; Zbl 0353.05009)]) which turn out to be those configurations having a coherent algebra of adjacency matrices. Each coherent algebra has an associated type which is a symmetric matrix of positive entries of order t where t is a number determined by a particular partition of the underlying coherent configuration. The author considers various types, paying particular attention to \(2\times 2\) and \(3\times 3\) types, showing how some of these correspond to designs considered by \textit{J. M. Goethals} and \textit{J. J. Seidel} [Can. J. Math. 22, 597-614 (1970; Zbl 0198.293)], \textit{H. Enemoto}, \textit{N. Ito} and \textit{R. Noda} [Osaka J. Math. 16, 39-43 (1979; Zbl 0401.05017)], \textit{P. J. Cameron} and \textit{J. H. van Lint} [Graphs, Codes and Designs (Lond. Math. Soc. Lect. Note Ser. 43 (1980; Zbl 0427.05001)], and others.
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 05B99
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 4002110
zbMATH Keywords: quasisymmetric designs
zbMATH Keywords: coherent configurations
reviewed by: John Clark
MaRDI profile type: Publication
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full work available at URL: s0024-3795(87)90326-0
OpenAlex ID: W4213243275
description: scientific article; zbMATH DE number 4002110