Coherent algebra
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.
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, 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.