Terwilliger Algebras of Wreath Powers of One-Class Association Schemes

TL;DR

This paper investigates the structure and properties of Terwilliger algebras of wreath powers of one-class association schemes, revealing their dimensions, regularity, and explicit algebraic isomorphisms.

Contribution

It provides a detailed analysis of the Terwilliger algebra structure for wreath powers of one-class association schemes, including explicit algebraic decompositions.

Findings

The wreath power schemes are triple-regular.

The dimension of the Terwilliger algebra is determined.

Explicit algebraic isomorphisms for the Terwilliger algebra are given.

Abstract

In this paper, we study the subconstituent algebras, also called as Terwilliger algebras, of association schemes that are obtained as the wreath product of one-class association schemes $K_n=H(1, n)$ for $n\ge 2$. We find that the $d$-class association scheme $K_{n_{1}}\wr K_{n_{2}} \wr ... \wr K_{n_{d}}$ formed by taking the wreath product of $K_{n_{i}}$ has the triple-regularity property. We determine the dimension of the Terwilliger algebra for the association scheme $K_{n_{1}}\wr K_{n_{2}}\wr ... \wr K_ {n_{d}}$. We give a description of the structure of the Terwilliger algebra for the wreath power $(K_n)^{\wr d}$ for $n \geq 2$ by studying its irreducible modules. In particular, we show that the Terwilliger algebra of $(K_n)^{\wr d}$ is isomorphic to $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C})^{\oplus \frac12d(d+1)}$ for $n\ge3$, and $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C})^{\oplus \frac12d(d-1)}$ for $n=2$.