Structure of Terwilliger algebras of quasi-thin association schemes

TL;DR

This paper proves that the Terwilliger algebra of a quasi-thin association scheme has a specific algebraic structure, enabling complete determination of many of its homological and representation-theoretic properties, especially over fields of characteristic 2.

Contribution

It establishes that these Terwilliger algebras are quasi-hereditary cellular algebras and describes their basic algebra structure, advancing understanding of their representation theory.

Findings

Terwilliger algebra is quasi-hereditary cellular algebra

Basic algebra is dual extension of a star with all arrows to the center in characteristic 2

Nakayama conjecture holds for these Terwilliger algebras

Abstract

We show that the Terwilliger algebra of a quasi-thin association scheme over a field is always a quasi-hereditary cellular algebra in the sense of Cline-Parshall-Scott and of Graham-Lehrer, repsectively, and that the basic algebra of the Terwilliger algebra is the dual extension of a star with all arrows pointing to its center if the field has characteristic $2$. Thus many homological and representation-theoretic properties of these Terwilliger algebras can be determined completely. For example, the Nakayama conjecture holds for Terwilliger algebras of quasi-thin association schemes.