Association schemes on triples from affine special semilinear groups

TL;DR

This paper explores association schemes on triples derived from the actions of affine special semilinear groups, providing new insights into their structure and parameters.

Contribution

It introduces a novel approach to constructing ASTs using semidirect products of ASL(k,n) with Galois subgroups, detailing their combinatorial parameters.

Findings

Determined sizes and third valencies of the ASTs.

Calculated intersection numbers for the ASTs.

Analyzed the group actions leading to these schemes.

Abstract

Association schemes on triples (ASTs) are 3-dimensional analogues of classical association schemes. If a group acts two-transitively on a set, the orbits of the action induced on the triple Cartesian product of that set yields an AST. By considering the actions of semidirect products of the affine special linear group ASL(k,n) with subgroups of the Galois group Gal(GF(n)), we obtain the sizes, third valencies, and intersection numbers of the ASTs obtained from subgroups of the affine special semilinear group.