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Coherent algebras. (English) Zbl 0618.05014
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 J. M. Goethals and J. J. Seidel [Can. J. Math. 22, 597-614 (1970; Zbl 0198.293)], H. Enemoto, N. Ito and R. Noda [Osaka J. Math. 16, 39-43 (1979; Zbl 0401.05017)], P. J. Cameron and J. H. van Lint [Graphs, Codes and Designs (Lond. Math. Soc. Lect. Note Ser. 43 (1980; Zbl 0427.05001)], and others.
Reviewer: J.Clark

MSC:
05B30 Other designs, configurations
05B99 Designs and configurations
15A30 Algebraic systems of matrices
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References:
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