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

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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

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  • 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

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  • 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

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References

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  1. ^ a b 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.