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Structure of Terwilliger algebras of quasi-thin association schemes

Zhenxian Chen, Changchang Xi

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.

Structure of Terwilliger algebras of quasi-thin association schemes

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 . 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.

Paper Structure

This paper contains 5 sections, 8 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Proof of the result
  3. Terwilliger algebras of association schemes
  4. Cellular and quasi-hereditary algebras
  5. Proof of the main result

Key Result

Theorem 1.1

Let $R$ be a field of arbitrary characteristic and $S$ a quasi-thin association scheme on a finite set. Then the Terwilliger $R$-algebra of $S$ is quasi-hereditary in the sense of Cline-Parshall-Scott, and cellular in the sense of Graham-Lehrer. Moreover, if the field $R$ has characteristic $2$, the

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 5 more