On the association scheme of perfect matchings and their designs

TL;DR

This paper explores the structure of perfect matchings and their designs within the association scheme derived from the Gelfand pair $(S_{2n},S_2 times S_n)$, unifying previous results and establishing new existence and non-existence theorems.

Contribution

It introduces a unified framework for perfect matchings and hyperfactorisations using association schemes from Gelfand pairs, extending prior work and providing new theoretical insights.

Findings

Unified framework for matchings and designs via association schemes

New existence and non-existence results for perfect matchings

Application of group algebra and representation theory methods

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 $S_{2n}$.