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 $W$ of order $n$ and rank $r$ is a matrix subalgebra of the full matrix algebra of $( n \times n )$-matrices over $\mathbf{C}$ such that:

$W$ is closed with respect to the Hermitian adjoint, which is defined by $A ^ { * } = ( a _ { i , j } ) ^ { * } = ( \overline { a _ { j , i } } )$ for $A = ( a _ { i ,\, j } ) \in W$;

$I \in W$, where $I$ is the unit matrix;

$J \in W$, where $J$ is the all-one matrix;

$W$ is closed with respect to Schur–Hadamard multiplication $\circ $, where $A \circ B = ( a _ { i , j} b _ { i , j} )$ for $A = ( a_{i ,\, j} )$, $B = ( b _ { i , j} )$, $A , B \in W$.

Each coherent algebra $W$ has a unique basis of zero-one matrices $\{ A _ { 1 } , \dots , A _ { r } \}$ such that:

1) $\sum _ { i = 1 } ^ { r } A _ { i } = J$;

2) $\forall 1 \leq i \leq r \exists 1 \leq j \leq r : A _ { i } ^ { T } = A _ { j }$, where $A _ { i } ^ { T }$ is the matrix transposed to $A_i$;

3) $A _ { i }\, A _ { j } = \sum _ { k = 1 } ^ { r } p _ { i ,\, j } ^ { k } \,A _ { k }$.

Property 1) implies that the basis $\{ A _ { 1 } , \dots , A _ { r } \}$ consists of mutually orthogonal idempotents with respect to the Schur–Hadamard product. This basis is called the standard basis of $W$. The non-negative integer structure constants $p _ { i,j } ^ { k }$ are important numerical invariants of $W$. The notation $W = \langle A _ { 1 } , \dots , A _ { r } \rangle$ indicates that $W$ is a coherent algebra with standard basis $\{ A _ { 1 } , \dots , A _ { r } \}$.

Let $X = \{ 1 , \dots , n \}$ and denote by $R_l = \{ ( i , j ) : a _ { i , j } = 1 \}$ a binary relation over $X$. $R_l$ is called the support of the zero-one matrix $A_{l} = ( a _ { i,j })$ (or, in other words, $A _ { l }$ is the adjacency matrix of the graph $\Gamma _ { l } = ( X , R _ { l } )$ with vertex set $X$ and set $R_l$ of directed edges). The system of relations $\mathfrak { M } = ( X , \{ R _ { i } \} _ { 1 \leq i \leq r } )$ obtained in this way from a coherent algebra $W$ is called a coherent configuration.

The structure constants $p _ { i,j } ^ { k }$ of $W$ are sometimes called the intersection numbers of $\mathfrak{M}$. They have the following combinatorial interpretation:

\begin{equation*} \forall ( x , y ) \in R _ { k }: \end{equation*}

\begin{equation*} p _ { i ,\, j } ^ { k } = | \{ z \in X : ( x , z ) \in R _{i}\, \& ( z , y ) \in R _ { j } \} |. \end{equation*}

A coherent configuration $\mathfrak{M}$ is called homogeneous if one of its basic relations, say $R _ { 1 }$, coincides with the diagonal relation $\Delta = \{ ( x , x ) : x \in X \}$. In terms of matrices, a coherent algebra $W = \langle A _ { 1 } , \dots , A _ { r } \rangle$ is called a Bose–Mesner algebra (briefly BM-algebra) if $A _ { 1 } = I$. 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 $\mathfrak { M } = ( X , \{ R _ { i } \} _ { 1 \leq i \leq r } )$ be a coherent configuration. A subset $\emptyset \neq M \subseteq X$ is called a fibre of $\mathfrak{M}$ if

\begin{equation} \tag{a1} \forall 1 \leq i \leq r : R _ { i } \subseteq M ^ { 2 } \vee R _ { i } \bigcap M ^ { 2 } = \emptyset \end{equation}

and $M$ 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 $\left( \begin{array} { l l } { 3 } & { 2 } \\ { 2 } & { 3 } \end{array} \right)$" 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)