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 $$I$$ and the all-ones matrix $$J$$.

Definitions
A subspace $$\mathcal{A}$$ of $$\mathrm{Mat}_{n \times n}(\mathbb{C})$$ is said to be a coherent algebra of order $$n$$ if: A coherent algebra $$\mathcal{A} $$ is said to be: The set $$\Gamma(\mathcal{A})$$ of Schur-primitive matrices in a coherent algebra $$\mathcal{A}$$ is defined as $$\Gamma(\mathcal{A}) := \{ M \in \mathcal{A} : M \circ M = M, M \circ N \in \operatorname{span} \{ M \} \text{ for all } N \in \mathcal{A} \} $$.
 * $$I, J \in \mathcal{A}$$.
 * $$M^{T} \in \mathcal{A}$$ for all $$M \in \mathcal{A}$$.
 * $$MN \in \mathcal{A}$$ and $$M \circ N \in \mathcal{A}$$ for all $$M, N \in \mathcal{A}$$.
 * Homogeneous if every matrix in $$\mathcal{A} $$ has a constant diagonal.
 * Commutative if $$\mathcal{A} $$ is commutative with respect to ordinary matrix multiplication.
 * Symmetric if every matrix in $$\mathcal{A} $$ is symmetric.

Dually, the set $$\Lambda(\mathcal{A})$$ of primitive matrices in a coherent algebra $$\mathcal{A}$$ is defined as $$\Lambda(\mathcal{A}) := \{ M \in \mathcal{A} : M^{2} = M, MN \in \operatorname{span} \{ M \} \text{ for all } N \in \mathcal{A} \} $$.

Examples

 * The centralizer of a group of permutation matrices is a coherent algebra, i.e. $$\mathcal{W}$$ is a coherent algebra of order $$n$$ if $$\mathcal{W} := \{ M \in \mathrm{Mat}_{n \times n}(\mathbb{C}) : MP = PM \text { for all } P \in S \}$$ for a group $$S$$ of $$n \times n$$ permutation matrices. Additionally, the centralizer of the group of permutation matrices representing the automorphism group of a graph $$G$$ is homogeneous if and only if $$G$$ 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. $$\mathcal{W} := \operatorname{span} \{ A(u,v) : u,v \in V \}$$ where $$A(u,v) \in \operatorname{Mat}_{V \times V}(\mathbb{C})$$ is defined as $$(A(u,v))_{x, y} := \begin{cases} 1 \ \text{if } (x, y) = (u^{g}, v^{g}) \text { for some } g \in G \\ 0 \text{ otherwise } \end{cases}$$for all $$u, v \in V$$ of a finite set $$V$$ acted on by a finite group $$G$$.
 * The span of a regular representation of a finite group as a group of permutation matrices over $$\mathbb{C}$$ is a coherent algebra.

Properties

 * The intersection of a set of coherent algebras of order $$n$$ is a coherent algebra.
 * The tensor product of coherent algebras is a coherent algebra, i.e. $$\mathcal{A} \otimes \mathcal{B} := \{ M \otimes N : M \in \mathcal{A} \text{ and } N \in \mathcal{B} \}$$ if $$\mathcal{A} \in \operatorname{Mat}_{m \times m}(\mathbb{C})$$ and $$\mathcal{B} \in \mathrm{Mat}_{n \times n}(\mathbb{C})$$ are coherent algebras.
 * The symmetrization $$\widehat{\mathcal{A}} := \operatorname{span} \{ M + M^{T} : M \in \mathcal{A} \}$$ of a commutative coherent algebra $$\mathcal{A}$$ is a coherent algebra.
 * If $$\mathcal{A}$$ is a coherent algebra, then $$M^{T} \in \Gamma(\mathcal{A})$$ for all $$M \in \mathcal{A}$$, $$\mathcal{A} = \operatorname{span} \left ( \Gamma(\mathcal{A} \right ))$$, and $$I \in \Gamma(\mathcal{A})$$ if $$\mathcal{A}$$ is homogeneous.
 * Dually, if $$\mathcal{A}$$ is a commutative coherent algebra (of order $$n$$), then $$E^{T}, E^{*} \in \Lambda(\mathcal{A})$$ for all $$E \in \mathcal{A}$$, $$\frac{1}{n} J \in \Lambda(\mathcal{A})$$, and $$\mathcal{A} = \operatorname{span} \left ( \Lambda(\mathcal{A} \right ))$$ 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.