On some $p$-transitive association schemes

TL;DR

This paper introduces the concept of p-transitive association schemes, generalizing group algebra modules, and classifies certain p-transitive schemes with specific structural properties.

Contribution

It defines p-transitive association schemes and provides a complete classification of p-transitive quasi-thin schemes and those with thin residue.

Findings

Classified p-transitive quasi-thin association schemes

Classified p-transitive schemes with thin thin residue

Established structural properties of p-transitive schemes

Abstract

In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique trivial submodule to the case of the regular modules of modular adjacency algebras. We completely determine the $p$-transitive quasi-thin association schemes and the $p$-transitive association schemes with thin thin residue by their structure theory properties.