Association schemes for diagonal groups

TL;DR

This paper constructs association schemes with diagonal groups as automorphisms, explores their properties, and characterizes AS-free groups, revealing connections to Latin squares and hypercubes.

Contribution

It introduces new association schemes for diagonal groups, analyzes their parameters, and characterizes AS-free groups as 2-homogeneous or almost simple.

Findings

Association schemes have ranks related to partitions of n.

For n=3, the scheme relates to Latin square graphs.

AS-free groups are either 2-homogeneous or almost simple.

Abstract

For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at most $|G|$ parts, so is $p(n)$ if $|G|\ge n$; its parameters depend only on $n$ and $|G|$. For $n=2$, the association scheme is trivial, while for $n=3$ its relations are the Latin square graph associated with the Cayley table of $G$ and its complement. A transitive permutation group $G$ is said to be \emph{AS-free} if there is no non-trivial association scheme admitting $G$ as a group of automorphisms. A consequence of our construction is that an AS-free group must be either $2$-homogeneous or almost simple. We construct another association scheme, finer than the above scheme if $n>3$, from the Latin hypercube consisting of $n$-tuples of elements of $G$ with product the identity.