Coherent algebra

Algebras introduced by D.G. Higman, first in relational language under the name coherent configuration [a4] and later in terms of matrices [a6]. The slightly different axiomatics of cellular algebras were independently suggested by B.Yu. Weisfeiler and A.A. Leman (cf. also Cellular algebra).

Like association schemes (cf. also Association scheme) and Bose–Mesner algebras, coherent algebras provide a wide and solid foundation for investigations in various areas of algebraic combinatorics.

A coherent algebra of order and rank is a matrix subalgebra of the full matrix algebra of -matrices over such that:

is closed with respect to the Hermitian adjoint, which is defined by for ;

, where is the unit matrix;

, where is the all-one matrix;

is closed with respect to Schur–Hadamard multiplication , where for , , . Each coherent algebra has a unique basis of zero-one matrices such that:

1) ;

2) , where is the matrix transposed to ;

3) . Property 1) implies that the basis consists of mutually orthogonal idempotents with respect to the Schur–Hadamard product. This basis is called the standard basis of . The non-negative integer structure constants are important numerical invariants of . The notation indicates that is a coherent algebra with standard basis .

Let and denote by a binary relation over . is called the support of the zero-one matrix (or, in other words, is the adjacency matrix of the graph with vertex set and set of directed edges). The system of relations obtained in this way from a coherent algebra is called a coherent configuration.

The structure constants of are sometimes called the intersection numbers of . They have the following combinatorial interpretation:

A coherent configuration is called homogeneous if one of its basic relations, say , coincides with the diagonal relation . In terms of matrices, a coherent algebra is called a Bose–Mesner algebra (briefly BM-algebra) if . Note that according to E. Bannai and T. Ito [a1], a homogeneous coherent configuration is also called an association scheme (not necessarily commutative; cf. also Association scheme).

Let be a coherent configuration. A subset is called a fibre of if

(a1)

and is a minimal (with respect to inclusion) subset satisfying condition (a1). The coherent algebras with one fibre are exactly the BM-algebras. Coherent algebras with few fibres may be used for a unified presentation and investigation of various combinatorial objects, see, for example, [a3], [a7], [a9].

An important class of coherent algebras consists of the centralizer algebras of permutation groups (not necessarily transitive) [a2], [a10] (cf. also Permutation group; Centralizer). This leads to many important applications of coherent algebras.

It was Higman [a5], [a8] who developed the foundations of the representation theory of coherent algebras as a generalization of the representation theory of finite permutation groups (cf. also Finite group, representation of a).

References

[a1]	E. Bannai, T. Ito, "Algebraic combinatorics" , I , Benjamin/Cummings (1984)
[a2]	I.A. Faradžev, M.H. Klin, M.E. Muzichuk, "Cellular rings and groups of automorphisms of graphs" I.A. Faradžev (ed.) et al. (ed.) , Investigations in Algebraic Theory of Combinatorial Objects , Kluwer Acad. Publ. (1994) pp. 1–152
[a3]	W.H. Haemers, D.G. Higman, "Strongly regular graphs with strongly regular decomposition" Linear Alg. & Its Appl. , 114/115 (1989) pp. 379–398
[a4]	D.G. Higman, "Coherent configurations I" Rend. Sem. Mat. Univ. Padova , 44 (1970) pp. 1–25
[a5]	D.G. Higman, "Coherent configurations, Part I: Ordinary representation theory" Geom. Dedicata , 4 (1975) pp. 1–32
[a6]	D.G. Higman, "Coherent algebras" Linear Alg. & Its Appl. , 93 (1987) pp. 209–239
[a7]	D.G. Higman, "Strongly regular designs and coherent configurations of type " Europ. J. Combin. , 9 (1988) pp. 411–422
[a8]	D.G. Higman, "Computations related to coherent configurations" Congr. Numer. , 75 (1990) pp. 9–20
[a9]	M.E. Muzychuk, M. Klin, "On graphs with three eigenvalues" Discr. Math. , 189 (1998) pp. 191–207
[a10]	H. Wielandt, "Finite permutation groups" , Acad. Press (1964)