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 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)