general.hpp File Reference

General plans regarding association schemes. More...

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Detailed Description

General plans regarding association schemes.

Connections to other modules
Other tools
  • R should have support for association schemes?
  • Hanaki and Miyamoto are classifying association scheme in GAP, so that there should be support in it.
Basic notions
  • Value associations (arbitrary matrices)
    1. Association classes
    2. Value incidence matrices
    3. Value adjacency matrices for square matrices
  • Association products (arbitrary and square matrices)
    1. The transposed association products for arbitrary matrices
    2. The association products for square matrices.
  • Parameters of the first kind ("intersection numbers")
    1. For square matrices only, but we consider general square matrices and symmetric matrices.
    2. Symmetry for undirected value-associated matrices
  • The problems of different levels of "equipment"
    1. Just as a matrix
    2. Together with value range, association classes and incidence matrices
    3. Additionally also with (transposed) association products
    4. With "neutral element"
    5. With parameters of the first kind
  • Tests for the various properties (only for square matrices)
    1. directed value-associated (the intersection number exist)
    2. undirected value-associated (additionally the matrix is symmetric)
    3. association scheme (undirected value-associated with "neutral element")
Basic examples
  • Trivial association schemes
  • Divisible association schemes
  • Rectangular association schemes
  • Triangular association schemes
  • Johnson association schemes
  • Association schemes of latin square type
  • Association schemes from distance-regular graphs
  • Hamming association schemes
  • Association schemes from groups

Definition in file general.hpp.