Coherent algebra

Definitions

• .
• for all.
• and for all.

• 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.

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, 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 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.