Terwilliger Algebras of Wreath Powers of One-Class Association Schemes

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 $n\ge 2$ by studying its irreducible modules. In particular, we show that the Terwilliger algebra of $(K_n{)}^{\wr d}$ is isomorphic to $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C}{)}^{\oplus \frac{1}{2}d(d+1)}$ for $n\ge 3$, and $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C}{)}^{\oplus \frac{1}{2}d(d-1)}$ for $n=2$.

27 pages