Английская Википедия:Coherent algebra
Шаблон:Short description Шаблон:Refimprove 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>I</math> and the all-ones matrix <math>J</math>.[1]
Definitions
A subspace <math>\mathcal{A}</math> of <math>\mathrm{Mat}_{n \times n}(\mathbb{C})</math> is said to be a coherent algebra of order <math>n</math> if:
- <math>I, J \in \mathcal{A}</math>.
- <math>M^{T} \in \mathcal{A}</math> for all <math>M \in \mathcal{A}</math>.
- <math>MN \in \mathcal{A}</math> and <math>M \circ N \in \mathcal{A}</math> for all <math>M, N \in \mathcal{A}</math>.
A coherent algebra <math>\mathcal{A} </math> is said to be:
- Homogeneous if every matrix in <math>\mathcal{A} </math> has a constant diagonal.
- Commutative if <math>\mathcal{A} </math> is commutative with respect to ordinary matrix multiplication.
- Symmetric if every matrix in <math>\mathcal{A} </math> is symmetric.
The set <math>\Gamma(\mathcal{A})</math> of Schur-primitive matrices in a coherent algebra <math>\mathcal{A}</math> is defined as <math>\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>\Lambda(\mathcal{A})</math> of primitive matrices in a coherent algebra <math>\mathcal{A}</math> is defined as <math>\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>\mathcal{W}</math> is a coherent algebra of order <math>n</math> if <math>\mathcal{W} := \{ M \in \mathrm{Mat}_{n \times n}(\mathbb{C}) : MP = PM \text { for all } P \in S \}</math> for a group <math>S</math> of <math>n \times n</math> permutation matrices. Additionally, the centralizer of the group of permutation matrices representing the automorphism group of a graph <math>G</math> is homogeneous if and only if <math>G</math> is vertex-transitive.[2]
- 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>\mathcal{W} := \operatorname{span} \{ A(u,v) : u,v \in V \}</math> where <math>A(u,v) \in \operatorname{Mat}_{V \times V}(\mathbb{C})</math> is defined as <math>(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>u, v \in V</math> of a finite set <math>V</math> acted on by a finite group <math>G</math>.
- The span of a regular representation of a finite group as a group of permutation matrices over <math>\mathbb{C}</math> is a coherent algebra.
Properties
- The intersection of a set of coherent algebras of order <math>n</math> is a coherent algebra.
- The tensor product of coherent algebras is a coherent algebra, i.e. <math>\mathcal{A} \otimes \mathcal{B} := \{ M \otimes N : M \in \mathcal{A} \text{ and } N \in \mathcal{B} \}</math> if <math>\mathcal{A} \in \operatorname{Mat}_{m \times m}(\mathbb{C})</math> and <math>\mathcal{B} \in \mathrm{Mat}_{n \times n}(\mathbb{C})</math> are coherent algebras.
- The symmetrization <math>\widehat{\mathcal{A}} := \operatorname{span} \{ M + M^{T} : M \in \mathcal{A} \}</math> of a commutative coherent algebra <math>\mathcal{A}</math> is a coherent algebra.
- If <math>\mathcal{A}</math> is a coherent algebra, then <math>M^{T} \in \Gamma(\mathcal{A})</math> for all <math>M \in \mathcal{A}</math>, <math>\mathcal{A} = \operatorname{span} \left ( \Gamma(\mathcal{A} \right ))</math>, and <math>I \in \Gamma(\mathcal{A})</math> if <math>\mathcal{A}</math> is homogeneous.
- Dually, if <math>\mathcal{A}</math> is a commutative coherent algebra (of order <math>n</math>), then <math>E^{T}, E^{*} \in \Lambda(\mathcal{A})</math> for all <math>E \in \mathcal{A}</math>, <math>\frac{1}{n} J \in \Lambda(\mathcal{A})</math>, and <math>\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.[1]
- 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
References
