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On the Terwilliger algebras of quasi-thin Schurian association schemes

Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

Abstract

We give necessary and sufficient conditions for the Terwilliger algebra of a quasi-thin Schurian association scheme to coincide with: (a) the centralizer algebra of a point stabilizer of its automorphism group, and (b) its subspace $T^0$. As a by-product, a full characterization of triply-transitive quasi-thin association schemes is given. Using the latter, we provide new infinite families of triply-transitive association schemes.

On the Terwilliger algebras of quasi-thin Schurian association schemes

Abstract

We give necessary and sufficient conditions for the Terwilliger algebra of a quasi-thin Schurian association scheme to coincide with: (a) the centralizer algebra of a point stabilizer of its automorphism group, and (b) its subspace . As a by-product, a full characterization of triply-transitive quasi-thin association schemes is given. Using the latter, we provide new infinite families of triply-transitive association schemes.
Paper Structure (18 sections, 17 theorems, 57 equations, 3 figures)

This paper contains 18 sections, 17 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background results
  3. Permutation group theory
  4. Algebras and vector spaces related to the Terwilliger algebra
  5. Thin and quasi-thin association schemes
  6. Triply-transitive association schemes
  7. Thin association schemes are triply transitive
  8. Diamond pairs
  9. Algebraic structure of the Terwilliger algebra
  10. Centralizer algebras of Schurian cases
  11. Centralizer algebras of non-Schurian cases
  12. Combinatorial structure of the Terwilliger algebra
  13. Triple regularity
  14. The Terwilliger algebra of non-triply-regular cases
  15. Triply-transitive quasi-thin schemes
  16. ...and 3 more sections

Key Result

Lemma 2.1

For any $v\in \Omega$, the dimension of $\operatorname{End}_{G_v}\left(V\right)$ is equal to the number of orbits of $G_v$ on $\Omega \times \Omega$.

Figures (3)

  • Figure 1: Configuration for diamond pairs.
  • Figure 2: Configuration for non-diamond pairs.
  • Figure 3: A diamond pair in $\mathfrak{X}$.

Theorems & Definitions (40)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 30 more