A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix and the all-ones matrix .
Definitions A subspace of is said to be a coherent algebra of order if: . for all . and for all . A coherent algebra is said to be: Homogeneous if every matrix in has a constant diagonal. Commutative if is commutative with respect to ordinary matrix multiplication. Symmetric if every matrix in is symmetric. The set of Schur-primitive matrices in a coherent algebra is defined as .
Dually, the set of primitive matrices in a coherent algebra is defined as .
Examples The centralizer of a group of permutation matrices is a coherent algebra, i.e. is a coherent algebra of order if for a group of permutation matrices. Additionally, the centralizer of the group of permutation matrices representing the automorphism group of a graph is homogeneous if and only if is vertex-transitive. The span of the set of matrices relating pairs of elements lying in the same orbit of a diagonal action of a finite group on a finite set is a coherent algebra, i.e. where is defined as for all of a finite set acted on by a finite group . The span of a regular representation of a finite group as a group of permutation matrices over is a coherent algebra.
Properties The intersection of a set of coherent algebras of order is a coherent algebra. The tensor product of coherent algebras is a coherent algebra, i.e. if and are coherent algebras. The symmetrization of a commutative coherent algebra is a coherent algebra. If is a coherent algebra, then for all , , and if is homogeneous. Dually, if is a commutative coherent algebra (of order ), then for all , , and as well. Every symmetric coherent algebra is commutative, and every commutative coherent algebra is homogeneous. A coherent algebra is commutative if and only if it is the Bose–Mesner algebra of a (commutative) association scheme. A coherent algebra forms a principal ideal ring under Schur product; moreover, a commutative coherent algebra forms a principal ideal ring under ordinary matrix multiplication as well.
See also Association scheme Bose–Mesner algebra
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