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

Jiu-Yang He, Yu Jiang

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.

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

Abstract

The Terwilliger algebras of association schemes over an arbitrary field were called the Terwilliger -algebras of association schemes in [9]. In this paper, we study the Terwilliger -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 -algebras of factorial association schemes. Moreover, we determine all Terwilliger -algebras of factorial association schemes that are the symmetric -algebras or the Frobenius -algebras.

Paper Structure

This paper contains 12 sections, 103 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Basic notation and preliminaries
  3. Conventions
  4. Schemes
  5. Algebras
  6. Terwilliger $\mathbb{F}$-algebras of schemes
  7. Algebraic structure of $\mathbb{T}$: $\mathbb{F}$-Basis
  8. Algebraic structure of $\mathbb{T}$: Center
  9. Algebraic structure of $\mathbb{T}$: Semisimplicity
  10. Algebraic structure of $\mathbb{T}$: Jacobson radical
  11. Algebraic structure of $\mathbb{T}$: Quotient $\mathbb{F}$-algebra
  12. Algebraic structure of $\mathbb{T}$: Wedderburn-Artin decomposition

Key Result

Lemma 2.1

Z Assume that $g\in\mathbb{N}$. Assume that $\mathfrak{S}$ is the direct product of the schemes $\mathfrak{S}_1, \mathfrak{S}_2, \ldots, \mathfrak{S}_g$ and $\mathbf{i}, \mathbf{j}, \mathbf{k}\in\mathbb{E}$. Then $p_{\mathbf{i},\mathbf{j}}^\mathbf{k}=\prod_{h=1}^g p_{\mathbf{i}_h,\mathbf{j}_h}^{\mat

Theorems & Definitions (201)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 191 more