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 $J$ .{{Cite web|url=http://www.math.uwaterloo.ca/~cgodsil/pdfs/assoc2.pdf|title=Association Schemes|last=Godsil|first=Chris|date=2010}}

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.{{Cite journal|last=Godsil|first=Chris|date=2011-01-26|title=Periodic Graphs|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p23|journal=The Electronic Journal of Combinatorics|volume=18|issue=1|pages=P23|issn=1077-8926|arxiv=0806.2074}}
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.

References

Category:Algebras

Category:Algebraic combinatorics

coherent algebra

Definitions

Examples

Properties

See also

References