Coherent Configuration

A coherent configuration on a finite set is a set partition , , ..., of . Each is a basis relation, meaning a binary relation that is one part of this set partition. The diagonal relation is a union of the . For every basis relation , the relation ${(y,x):(x,y) in R_i}$ is another . Furthermore, whenever , the number

$p_(ij)^k=|{z:(x,z) in R_i and (z,y) in R_j}|$

depends only on , , and . The constants are the intersection numbers of the coherent configuration.

A coherent configuration is called homogeneous when the diagonal relation itself is one basis relation; homogeneous coherent configurations are association schemes. Its adjacency matrices span an adjacency algebra, which for a homogeneous coherent configuration is the Bose-Mesner algebra. A coherent configuration is called Schurian when its basis relations are the orbitals under the componentwise group action on induced by a permutation group on .

See also

Adjacency Algebra, Association Scheme, Association Scheme Intersection Number, Basis Relation, Bose-Mesner Algebra, Diagonal Relation, Homogeneous Coherent Configuration, Permutation Group, Relation, Schurian Coherent Configuration, Schurian Graph, Schurian Scheme, Set Partition, Weisfeiler-Leman Algorithm, Weisfeiler-Leman Dimension

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References

Higman, D. G. "Coherent Configurations. I. Ordinary Representation Theory." Geom. Dedicata 4, 1-32, 1975. https://doi.org/10.1007/BF00147398.

Cite this as:

Weisstein, Eric W. "Coherent Configuration." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CoherentConfiguration.html

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