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:

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

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

• 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.

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

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.

See also

• Association scheme
• Bose–Mesner algebra