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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.[2]
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 forms a principal ideal ring under Schur product; moreover, a commutative coherent algebra forms a principal ideal ring under ordinary matrix multiplication as well.