## 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

### Keywords:

quasisymmetric designs; coherent configurations
Full Text:

### References:

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