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 ${\displaystyle I}$ and the all-ones matrix ${\displaystyle J}$.^[1]

Definitions

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

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

A coherent algebra ${\displaystyle {\mathcal {A}}}$ is said to be:

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

The set ${\displaystyle \Gamma ({\mathcal {A}})}$ of Schur-primitive matrices in a coherent algebra ${\displaystyle {\mathcal {A}}}$ is defined as ${\displaystyle \Gamma ({\mathcal {A}}):=\{M\in {\mathcal {A}}:M\circ M=M,M\circ N\in \operatorname {span} \{M\}{\text{ for all }}N\in {\mathcal {A}}\}}$.

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

Examples

• The centralizer of a group of permutation matrices is a coherent algebra, i.e. ${\displaystyle {\mathcal {W}}}$ is a coherent algebra of order ${\displaystyle n}$ if ${\displaystyle {\mathcal {W}}:=\{M\in \mathrm {Mat} _{n\times n}(\mathbb {C} ):MP=PM{\text{ for all }}P\in S\}}$ for a group ${\displaystyle S}$ of ${\displaystyle n\times n}$ permutation matrices. Additionally, the centralizer of the group of permutation matrices representing the automorphism group of a graph ${\displaystyle G}$ is homogeneous if and only if ${\displaystyle G}$ 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. ${\displaystyle {\mathcal {W}}:=\operatorname {span} \{A(u,v):u,v\in V\}}$ where ${\displaystyle A(u,v)\in \operatorname {Mat} _{V\times V}(\mathbb {C} )}$ is defined as ${\displaystyle (A(u,v))_{x,y}:={\begin{cases}1\ {\text{if }}(x,y)=(u^{g},v^{g}){\text{ for some }}g\in G\\0{\text{ otherwise }}\end{cases}}}$for all ${\displaystyle u,v\in V}$ of a finite set ${\displaystyle V}$ acted on by a finite group ${\displaystyle G}$.
• The span of a regular representation of a finite group as a group of permutation matrices over ${\displaystyle \mathbb {C} }$ is a coherent algebra.

Properties

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

• Association scheme
• Bose–Mesner algebra

References

1. Godsil, Chris (2010). "Association Schemes" (PDF).
2. Godsil, Chris (2011-01-26). "Periodic Graphs". The Electronic Journal of Combinatorics. 18 (1): P23. arXiv:0806.2074. ISSN 1077-8926.