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On the association scheme of perfect matchings and their designs

John Bamberg, Lukas Klawuhn

Abstract

We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$ and using the representation theory of $S_{2n}$.

On the association scheme of perfect matchings and their designs

Abstract

We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph . We show that they are special subsets of the association scheme obtained from the Gelfand pair . This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra and using the representation theory of .

Paper Structure

This paper contains 15 sections, 15 theorems, 56 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions
  4. Characters
  5. Character Theory of the Symmetric Group
  6. Association Schemes and Gelfand Pairs
  7. Association schemes
  8. Group algebras
  9. Gelfand Pairs
  10. The perfect matching association scheme
  11. Eigenvalues of the perfect matching association scheme
  12. Designs
  13. Antidesigns
  14. Examples for small $n$
  15. Open questions

Key Result

Theorem 1.1

A non-empty set of perfect matchings is a $\lambda$-factorisation if and only if its dual degree set contains no partition $\mu\vdash n$ with $\lambda \trianglelefteqslant \mu \neq (n)$.

Figures (1)

  • Figure 1: Two perfect matchings and their union.

Theorems & Definitions (38)

  • Theorem 1.1: Paraphrase of Theorem \ref{['theorem:main1']}
  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2: cf. Mac1995
  • Theorem 3.3: GodMea2016
  • Corollary 3.4
  • proof
  • Definition 4.1
  • Example 4.2
  • Example 4.3
  • ...and 28 more