On some p-transitive association 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 principal submodule to the case of the regular modules of modular adjacency algebras of association schemes. We characterize the p-transitive quasi-thin association schemes and the p-transitive association schemes with thin thin residue by their structure theory properties. We also get some results with independent interests.

Keywords:

• Association scheme
• quasi-thin scheme
• scheme with thin thin residue

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05E30

Acknowledgments

The author thanks his former Ph.D. supervisor Dr. Kay Jin Lim for organizing the seminar on the theory of association schemes, where the main results of this paper were motivated and obtained. He also thanks Dr. Kay Jin Lim and Prof. Gang Chen for encouraging him to learn the theory of association schemes. Moreover, he gratefully thanks Prof. Akihide Hanaki for some helpful comments on an earlier version of this paper. Finally, he gratefully thanks an anonymous referee for his or her insightful comments on the submitted version of this paper.