Coherent algebra

Short description: Algebra of complex square matrices

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 [math]\displaystyle{ I }[/math] and the all-ones matrix [math]\displaystyle{ J }[/math].

Definitions

A subspace [math]\displaystyle{ \mathcal{A} }[/math] of [math]\displaystyle{ \mathrm{Mat}_{n \times n}(\mathbb{C}) }[/math] is said to be a coherent algebra of order [math]\displaystyle{ n }[/math] if:

• [math]\displaystyle{ I, J \in \mathcal{A} }[/math].
• [math]\displaystyle{ M^{T} \in \mathcal{A} }[/math] for all [math]\displaystyle{ M \in \mathcal{A} }[/math].
• [math]\displaystyle{ MN \in \mathcal{A} }[/math] and [math]\displaystyle{ M \circ N \in \mathcal{A} }[/math] for all [math]\displaystyle{ M, N \in \mathcal{A} }[/math].

A coherent algebra [math]\displaystyle{ \mathcal{A} }[/math] is said to be:

• Homogeneous if every matrix in [math]\displaystyle{ \mathcal{A} }[/math] has a constant diagonal.
• Commutative if [math]\displaystyle{ \mathcal{A} }[/math] is commutative with respect to ordinary matrix multiplication.
• Symmetric if every matrix in [math]\displaystyle{ \mathcal{A} }[/math] is symmetric.

The set [math]\displaystyle{ \Gamma(\mathcal{A}) }[/math] of Schur-primitive matrices in a coherent algebra [math]\displaystyle{ \mathcal{A} }[/math] is defined as [math]\displaystyle{ \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} \} }[/math].

Dually, the set [math]\displaystyle{ \Lambda(\mathcal{A}) }[/math] of primitive matrices in a coherent algebra [math]\displaystyle{ \mathcal{A} }[/math] is defined as [math]\displaystyle{ \Lambda(\mathcal{A}) := \{ M \in \mathcal{A} : M^{2} = M, MN \in \operatorname{span} \{ M \} \text{ for all } N \in \mathcal{A} \} }[/math].

Examples

• The centralizer of a group of permutation matrices is a coherent algebra, i.e. [math]\displaystyle{ \mathcal{W} }[/math] is a coherent algebra of order [math]\displaystyle{ n }[/math] if [math]\displaystyle{ \mathcal{W} := \{ M \in \mathrm{Mat}_{n \times n}(\mathbb{C}) : MP = PM \text { for all } P \in S \} }[/math] for a group [math]\displaystyle{ S }[/math] of [math]\displaystyle{ n \times n }[/math] permutation matrices. Additionally, the centralizer of the group of permutation matrices representing the automorphism group of a graph [math]\displaystyle{ G }[/math] is homogeneous if and only if [math]\displaystyle{ G }[/math] 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. [math]\displaystyle{ \mathcal{W} := \operatorname{span} \{ A(u,v) : u,v \in V \} }[/math] where [math]\displaystyle{ A(u,v) \in \operatorname{Mat}_{V \times V}(\mathbb{C}) }[/math] is defined as [math]\displaystyle{ (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} }[/math]for all [math]\displaystyle{ u, v \in V }[/math] of a finite set [math]\displaystyle{ V }[/math] acted on by a finite group [math]\displaystyle{ G }[/math].
• The span of a regular representation of a finite group as a group of permutation matrices over [math]\displaystyle{ \mathbb{C} }[/math] is a coherent algebra.

Properties

• The intersection of a set of coherent algebras of order [math]\displaystyle{ n }[/math] is a coherent algebra.
• The tensor product of coherent algebras is a coherent algebra, i.e. [math]\displaystyle{ \mathcal{A} \otimes \mathcal{B} := \{ M \otimes N : M \in \mathcal{A} \text{ and } N \in \mathcal{B} \} }[/math] if [math]\displaystyle{ \mathcal{A} \in \operatorname{Mat}_{m \times m}(\mathbb{C}) }[/math] and [math]\displaystyle{ \mathcal{B} \in \mathrm{Mat}_{n \times n}(\mathbb{C}) }[/math] are coherent algebras.
• The symmetrization [math]\displaystyle{ \widehat{\mathcal{A}} := \operatorname{span} \{ M + M^{T} : M \in \mathcal{A} \} }[/math] of a commutative coherent algebra [math]\displaystyle{ \mathcal{A} }[/math] is a coherent algebra.
• If [math]\displaystyle{ \mathcal{A} }[/math] is a coherent algebra, then [math]\displaystyle{ M^{T} \in \Gamma(\mathcal{A}) }[/math] for all [math]\displaystyle{ M \in \mathcal{A} }[/math], [math]\displaystyle{ \mathcal{A} = \operatorname{span} \left ( \Gamma(\mathcal{A} \right )) }[/math], and [math]\displaystyle{ I \in \Gamma(\mathcal{A}) }[/math] if [math]\displaystyle{ \mathcal{A} }[/math] is homogeneous.
• Dually, if [math]\displaystyle{ \mathcal{A} }[/math] is a commutative coherent algebra (of order [math]\displaystyle{ n }[/math]), then [math]\displaystyle{ E^{T}, E^{*} \in \Lambda(\mathcal{A}) }[/math] for all [math]\displaystyle{ E \in \mathcal{A} }[/math], [math]\displaystyle{ \frac{1}{n} J \in \Lambda(\mathcal{A}) }[/math], and [math]\displaystyle{ \mathcal{A} = \operatorname{span} \left ( \Lambda(\mathcal{A} \right )) }[/math] 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