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 and the all-ones matrix .[1]

Definitions

A subspace of is said to be a coherent algebra of order if:

  • .
  • for all .
  • and for all .

A coherent algebra is said to be:

  • Homogeneous if every matrix in has a constant diagonal.
  • Commutative if is commutative with respect to ordinary matrix multiplication.
  • Symmetric if every matrix in is symmetric.

The set of Schur-primitive matrices in a coherent algebra is defined as .

Dually, the set of primitive matrices in a coherent algebra is defined as .

Examples

  • The centralizer of a group of permutation matrices is a coherent algebra, i.e. is a coherent algebra of order if for a group of permutation matrices. Additionally, the centralizer of the group of permutation matrices representing the automorphism group of a graph is homogeneous if and only if 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. where is defined as for all of a finite set acted on by a finite group .
  • The span of a regular representation of a finite group as a group of permutation matrices over is a coherent algebra.

Properties

  • The intersection of a set of coherent algebras of order is a coherent algebra.
  • The tensor product of coherent algebras is a coherent algebra, i.e. if and are coherent algebras.
  • The symmetrization of a commutative coherent algebra is a coherent algebra.
  • If is a coherent algebra, then for all , , and if is homogeneous.
  • Dually, if is a commutative coherent algebra (of order ), then for all , , and 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

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