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

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 [8]. In this paper, we study the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We determine the $\mathbb{F}$-dimensions, the centers, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes.

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 [8]. In this paper, we study the Terwilliger -algebras of factorial association schemes. We determine the -dimensions, the centers, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger -algebras of factorial association schemes.
Paper Structure (13 sections, 102 theorems, 26 equations)

This paper contains 13 sections, 102 theorems, 26 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. Closed subsets of factorial schemes
  8. $\mathbb{F}$-Dimensions of Terwilliger $\mathbb{F}$-algebras of factorial schemes
  9. Centers of Terwilliger $\mathbb{F}$-algebras of factorial schemes
  10. Semisimplicity of Terwilliger $\mathbb{F}$-algebras of factorial schemes
  11. Jacobson radicals of Terwilliger $\mathbb{F}$-algebras of factorial schemes
  12. Structures of Terwilliger $\mathbb{F}$-algebras of factorial schemes I
  13. Structures of Terwilliger $\mathbb{F}$-algebras of factorial schemes II

Key Result

Lemma 2.1

Z Assume that $g, h, i\in [0,d]$. Then $k_gp_{hi}^g=k_hp_{gi'}^h=k_ip_{h'g}^i$. Moreover, $p_{hi}^g\neq 0$, $p_{gi'}^h\neq 0$, $p_{h'g}^i\neq 0$ are pairwise equivalent.

Theorems & Definitions (202)

  • 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 192 more