Families of Association Schemes on Triples from Two-Transitive Groups

TL;DR

This paper studies association schemes on triples derived from two-transitive groups, determining their sizes, valencies, and intersection numbers, revealing that many are commutative, thus extending the understanding of algebraic structures from group actions.

Contribution

It provides explicit parameters and intersection numbers for ASTs from various two-transitive groups, including classical, sporadic, and subgroup actions, highlighting their commutative properties.

Findings

ASTs from these groups have explicitly determined sizes and third valencies.

Intersection numbers are computed for ASTs from specific subgroup actions.

Many ASTs from projective and sporadic groups are shown to be commutative.

Abstract

Association schemes on triples (ASTs) are ternary analogues of classical association schemes. Analogous to Schurian association schemes, ASTs arise from the actions of two-transitive groups. In this paper, we obtain the sizes and third valencies of the ASTs obtained from the two-transitive permutation groups by determining the orbits of the groups' two-point stabilizers. Specifically, we obtain these parameters for the ASTs obtained from the actions of $S_n$ and $A_n$, $PGU(3,q)$, $PSU(3,q)$, and $Sp(2k,2)$, $Sz(2^{2k+1})$ and $Ree(3^{2k+1})$, some subgroups of $A\Gamma L(k,n)$, some subgroups of $P\Gamma L(k,n)$, and the sporadic two-transitive groups. Further, we obtain the intersection numbers for the ASTs obtained from these subgroups of $P\Gamma L(k,n)$ and $A \Gamma L(k,n)$, and the sporadic two-transitive groups. In particular, the ASTs from these projective and sporadic groups are commutative.