The Terwilliger algebras of the group association schemes of non-abelian finite groups admitting an abelian subgroup of index 2

TL;DR

This work determines the dimension and detailed algebraic structure of the Terwilliger algebras for group association schemes arising from non-abelian finite groups that contain an abelian subgroup of index $2$. By establishing that the relevant groups are triply transitive, the authors show $\mathcal{T}(G)=\widetilde{\mathcal{T}}(G)$ and derive the key dimension formula $\dim_{\mathbb{C}}\mathcal{T}(D_2)=\tfrac{1}{2}(3nd+n^2+4d^2)$ with $n=|A|$ and $d=\prod d_i$, along with case analyses for generalized dihedral, generalized dicyclic, and cyclic-index-$2$ variants. A complete Wedderburn decomposition of $\mathcal{T}(D_2)$ is provided, expressing $\mathcal{T}(D_2)$ as a direct sum of matrix algebras corresponding to $2d$ one-dimensional and $(n-d)/2$ two-dimensional irreducible representations, with explicit dimension formulas for the simple components in terms of the group and automorphism data. These results extend prior classifications for metacyclic-type groups and illuminate the representation-theoretic structure underlying the Terwilliger algebras of this natural non-abelian class.

Abstract

In this paper, we determine the dimension of the Terwilliger algebras of non-abelian finite groups admitting an abelian subgroup of index 2 by showing that they are triply transitive. Moreover, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of these groups.

Theorems & Definitions (20)

• Definition 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8