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Characterizations of amorphic schemes and fusions of pairs

Edwin R. van Dam, Jack H. Koolen, Yanzhen Xiong

TL;DR

This work defines the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses, and shows that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic.

Abstract

An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.

Characterizations of amorphic schemes and fusions of pairs

TL;DR

This work defines the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses, and shows that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic.

Abstract

An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.
Paper Structure (15 sections, 19 theorems, 17 equations)

This paper contains 15 sections, 19 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Association schemes
  4. Fusions
  5. Strongly regular graphs
  6. Amorphic schemes
  7. All pairs
  8. Schemes with many (negative) Latin square type relations
  9. Fusion and contraction in the fusing-relations graph
  10. Not a path
  11. Dualization
  12. Strongly regular idempotents
  13. Latin square type idempotents
  14. Main dual results
  15. Final remarks and problems

Key Result

Lemma 2.1

Let $M$ be one of the eigenmatrices of an association scheme. Let $t \geq 2$. For any $t$ rows of the principal part of $M$, there are at least $t$ columns of the principal part of $M$ that are not constant on the $t$ rows.

Theorems & Definitions (36)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 26 more