Coherent algebra

Template:Short description Template:Refimprove 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 ${\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times n}(ℂ)$ 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 $𝒲:=\{M\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times n}(ℂ):MP=PM\text{ for all }P\in S\}$ 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 $A(u,v)\in {\mathrm{Mat}}_{V\times V}(ℂ)$ is defined as $(A(u,v){)}_{x,y}:=\{\begin{array}{c}1\text{if }(x,y)=(u^g,v^g)\text{ for some }g\in G\\ 0\text{ otherwise }\end{array}$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 $ℂ$ 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 $𝒜\in {\mathrm{Mat}}_{m\times m}(ℂ)$ and $ℬ\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times n}(ℂ)$ 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