Coherent algebra

Short description: Algebra of complex square matrices

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 ${𝒜}$ of ${\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times n}(\mathbb{ℂ})$ is said to be a coherent algebra of order $n$ if:

• $I,J\in {𝒜}$.
• $M^T\in {𝒜}$ for all $M\in {𝒜}$.
• $MN\in {𝒜}$ and $M\circ N\in {𝒜}$ for all $M,N\in {𝒜}$.

A coherent algebra ${𝒜}$ is said to be:

• Homogeneous if every matrix in ${𝒜}$ has a constant diagonal.
• Commutative if ${𝒜}$ is commutative with respect to ordinary matrix multiplication.
• Symmetric if every matrix in ${𝒜}$ is symmetric.

The set $\mathrm{\Gamma }({𝒜})$ of Schur-primitive matrices in a coherent algebra ${𝒜}$ is defined as $\mathrm{\Gamma }({𝒜}):=\{M\in {𝒜}:M\circ M=M,M\circ N\in \mathrm{span}\{M\}\text{ for all }N\in {𝒜}\}$.

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

Examples

• The centralizer of a group of permutation matrices is a coherent algebra, i.e. ${𝒲}$ is a coherent algebra of order $n$ if ${𝒲}:=\{M\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times n}(\mathbb{ℂ}):MP=PM\text{ for all }P\in S\}$ 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. ${𝒲}:=\mathrm{span}\{A(u,v):u,v\in V\}$ where $A(u,v)\in {\mathrm{Mat}}_{V\times V}(\mathbb{ℂ})$ is defined as $(A(u,v){)}_{x,y}:=\{\begin{array}{l}1\text{if }(x,y)=(u^g,v^g)\text{ for some }g\in G\\ 0\text{ otherwise }\end{array}$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. ${𝒜}\otimes {ℬ}:=\{M\otimes N:M\in {𝒜}\text{ and }N\in {ℬ}\}$ if ${𝒜}\in {\mathrm{Mat}}_{m\times m}(\mathbb{ℂ})$ and ${ℬ}\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times n}(\mathbb{ℂ})$ are coherent algebras.
• The symmetrization $\widehat{{𝒜}}:=\mathrm{span}\{M+M^T:M\in {𝒜}\}$ of a commutative coherent algebra ${𝒜}$ is a coherent algebra.
• If ${𝒜}$ is a coherent algebra, then $M^T\in \mathrm{\Gamma }({𝒜})$ for all $M\in {𝒜}$, ${𝒜}=\mathrm{span}(\mathrm{\Gamma }({𝒜}))$, and $I\in \mathrm{\Gamma }({𝒜})$ if ${𝒜}$ is homogeneous.
• Dually, if ${𝒜}$ is a commutative coherent algebra (of order $n$), then $E^T,E^{*}\in \mathrm{\Lambda }({𝒜})$ for all $E\in {𝒜}$, $\frac{1}{n}J\in \mathrm{\Lambda }({𝒜})$, and ${𝒜}=\mathrm{span}(\mathrm{\Lambda }({𝒜}))$ 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