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 $\mathcal$ of $\mathrm_(\mathbb)$ is said to be a coherent algebra of order $n$ if:
* $I, J \in \mathcal$.
* $M^ \in \mathcal$ for all $M \in \mathcal$.
* $MN \in \mathcal$ and $M \circ N \in \mathcal$ for all $M, N \in \mathcal$.
A coherent algebra $\mathcal$ is said to be:
* ''Homogeneous'' if every matrix in $\mathcal$ has a constant diagonal.
* ''Commutative'' if $\mathcal$ is commutative with respect to ordinary matrix multiplication.
* ''Symmetric'' if every matrix in $\mathcal$ is symmetric.
The set $\Gamma(\mathcal)$ of ''Schur-primitive matrices'' in a coherent algebra $\mathcal$ is defined as $\Gamma(\mathcal) := \$.

Dually, the set $\Lambda(\mathcal)$ of ''primitive matrices'' in a coherent algebra $\mathcal$ is defined as $\Lambda(\mathcal) := \$.

Examples

* The centralizer of a group of permutation matrices is a coherent algebra, i.e. $\mathcal$ is a coherent algebra of order $n$ if $\mathcal := \$ 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 := \operatorname \$ where $A(u,v) \in \operatorname_(\mathbb)$ is defined as $(A(u,v))_ := \begin 1 \ \text (x, y) = (u^, v^) \text g \in G \\ 0 \text \end$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 $\mathbb$ 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 \otimes \mathcal := \$ if $\mathcal \in \operatorname_(\mathbb)$ and $\mathcal \in \mathrm_(\mathbb)$ are coherent algebras.
* The ''symmetrization'' $\widehat := \operatorname \$ of a commutative coherent algebra $\mathcal$ is a coherent algebra.
* If $\mathcal$ is a coherent algebra, then $M^ \in \Gamma(\mathcal)$ for all $M \in \mathcal$, $\mathcal = \operatorname \left ( \Gamma(\mathcal \right ))$, and $I \in \Gamma(\mathcal)$ if $\mathcal$ is homogeneous.
* Dually, if $\mathcal$ is a commutative coherent algebra (of order $n$), then $E^, E^ \in \Lambda(\mathcal)$ for all $E \in \mathcal$, $\frac J \in \Lambda(\mathcal)$, and $\mathcal = \operatorname \left ( \Lambda(\mathcal{A} \right ))$ 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

References

Algebras
Algebraic combinatorics