On Terwilliger $\mathbb{F}$-algebras of factorial association schemes

TL;DR

This paper investigates the structure of Terwilliger $ ext{F}$-algebras associated with factorial association schemes, providing detailed algebraic properties and classifications over arbitrary fields.

Contribution

It characterizes the centers, semisimplicity, radicals, and decompositions of these algebras, and classifies those that are symmetric or Frobenius over any field.

Findings

Determined centers and radicals of the algebras.

Established Wedderburn-Artin decompositions.

Classified symmetric and Frobenius cases.

Abstract

The Terwilliger algebras of association schemes over an arbitrary field $\mathbb{F}$ were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [9]. In this paper, we study the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We determine the centers, the semisimplicity, the Jacobson radicals and their nilpotent indices, the Wedderburn-Artin decompositions of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. Moreover, we determine all Terwilliger $\mathbb{F}$-algebras of factorial association schemes that are the symmetric $\mathbb{F}$-algebras or the Frobenius $\mathbb{F}$-algebras.