Coherent algebra explained

I and the all-ones matrix J .^[1]

Definitions

A subspace

l{A} of Mat _{n} ( C ) is said to be a coherent algebra of order n if:I, J \in l{A} .M ^{T} \in l{A} for all M \in l{A} .MN \in l{A} and M \circ N \in l{A} for all M, N \in l{A} .A coherent algebra l{A} is said to be:

• Homogeneous if every matrix in

l{A} has a constant diagonal.

• Commutative if

l{A} is commutative with respect to ordinary matrix multiplication.

• Symmetric if every matrix in

l{A} is symmetric.The set \Gamma( l{A}) of Schur-primitive matrices in a coherent algebra l{A} is defined as \Gamma( l{A}) : = \{ M \in l{A} : M \circ M = M, M \circ N \in \operatorname{span} \{ M \} forall N \in l{A} \} .

Dually, the set

Λ ( l{A}) of primitive matrices in a coherent algebra l{A} is defined as Λ ( l{A}) : = \{ M \in l{A} : M ^{2} = M, MN \in \operatorname{span} \{ M \} forall N \in l{A} \} .

Examples

• The centralizer of a group of permutation matrices is a coherent algebra, i.e.

l{W} is a coherent algebra of order n if l{W} : = \{ M \in Mat _{n} ( C ) : MP = PM forall P \in S \} for a group S of n x 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.^[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.

l{W} : = \operatorname{span} \{ A(u,v) : u,v \in V \} where A(u,v) \in \operatorname{Mat} _{V} ( C ) is defined as (A(u,v)) _{x,} : = \begin{cases} 1 if (x, y) = (u ^{g} , v ^{g} ) forsome g \in G \ 0 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

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.

l{A} ⊗ l{B} : = \{ M ⊗ N : M \in l{A} and N \in l{B} \} if l{A} \in \operatorname{Mat} _{m} ( C ) and l{B} \in Mat _{n} ( C ) are coherent algebras.

• The symmetrization

\widehat{ l{A}} : = \operatorname{span} \{ M + M ^{T} : M \in l{A} \} of a commutative coherent algebra l{A} is a coherent algebra.

• If

l{A} is a coherent algebra, then M ^{T} \in \Gamma( l{A}) for all M \in l{A} , l{A} = \operatorname{span} \left ( \Gamma( l{A} \right )) , and I \in \Gamma( l{A}) if l{A} is homogeneous.

• Dually, if

l{A} is a commutative coherent algebra (of order n ), then E ^{T} , E ^{*} \in Λ ( l{A}) for all E \in l{A} , (1)/(n) J \in Λ ( l{A}) , and l{A} = \operatorname{span} \left ( Λ ( l{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

1. Web site: Association Schemes. Godsil. Chris. 2010.
2. Godsil. Chris. 2011-01-26. Periodic Graphs. The Electronic Journal of Combinatorics. 18. 1. P23. 1077-8926. 0806.2074.