The structure of imprimitive non-symmetric 3-class association schemes
type: Item
instance of: scholarly article
title: The structure of imprimitive non-symmetric 3-class association schemes
zbMATH Open document ID: 0843.05102
author: R. W. Goldbach
author: H. L. Claasen
published in: European Journal of Combinatorics
publication date: 1996-08-13T00:00:00Z
review text: The authors begin to characterize the imprimitivity of non-symmetric 3-schemes (3-class association schemes) in terms of their parameters. The symmetrization of a non-symmetric 3-scheme is a symmetric 2-scheme. Let \({\mathbf R}= \{R_0,\dots, R_n\}\) be a family of binary relations on \(X\), and \((X, {\mathbf R})\) be a commutative \(n\)-scheme. Then \[ p^k_{ij}= |\{z\in X\mid (x, z)\in R_i\text{ and } (z, y)\in R_j\}| \] are independent of the choice of \((x, y)\in R_k\), and \(p^k_{ij}= p^k_{ji}\). Further, for \(R_i\in {\mathbf R}\) the inverse \(R^{- 1}_i= \{(y, x)\mid (x, y)\in R_i\}\) of the relation \(R_i\) belongs to \({\mathbf R}\) (the index of relation \(R^{- 1}_i\) is denoted by \(i^R\)). Set \(v_i= p^0_{ii_R}\) and \(u= v_1/v_2\). Theorem. For an imprimitive non-symmetric 3-scheme \((X, {\mathbf R})\), there is only one of the following possibilities: (1) \(p^1_{13}= p^1_{23}\); (2) \(up^1_{33}= p^1_{23}\); and (3) \(p^1_{11}= p^2_{11}\). There are obvious ways to construct all the schemes of types (1), (2) (Theorems 5.5 and 7.1), connected with skew-Hadamard matrices. Only partial results are found in case (3).
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 847628
zbMATH Keywords: imprimitivity of non-symmetric 3-schemes
zbMATH Keywords: association scheme
zbMATH Keywords: symmetrization
zbMATH Keywords: skew-Hadamard matrices
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: eujc.1996.0003
OpenAlex ID: W2033097463
DOI: 10.1006/EUJC.1996.0003
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description: scientific article; zbMATH DE number 847628