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

$I, J \in 𝒜$ .
$M^{T} \in 𝒜$ for all $M \in 𝒜$ .
$M N \in 𝒜$ 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 $Γ (𝒜)$ of Schur-primitive matrices in a coherent algebra $𝒜$ is defined as $Γ (𝒜) := {M \in 𝒜 : M \circ M = M, M \circ N \in span {M} for all N \in 𝒜}$ .

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

Examples

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

References

1 2 Godsil, Chris (2010). "Association Schemes" (PDF).
↑ Godsil, Chris (2011-01-26). "Periodic Graphs". The Electronic Journal of Combinatorics. 18 (1): P23. arXiv:0806.2074. ISSN 1077-8926.

[:0-1] 1 2 Godsil, Chris (2010). "Association Schemes" (PDF).

[2] Godsil, Chris (2011-01-26). "Periodic Graphs". The Electronic Journal of Combinatorics. 18 (1): P23. arXiv:0806.2074. ISSN 1077-8926.

Coherent algebra

Definitions

Examples

Properties

See also

References