The Terwilliger algebra of the halved folded 2n-cube from the viewpoint of its automorphism group action

Abstract

Let $\frac{1}{2}{\overline{H}}(2n,2)$ denote the halved folded 2n-cube with vertex set X and let $T{:}{=}T(x)$ denote the Terwilliger algebra of $\frac{1}{2}{\overline{H}}(2n,2)$ with respect to a fixed vertex x. In this paper, we assume $n\ge 4$ and show that T coincides with the centralizer algebra of the stabilizer of x in the automorphism group of $\frac{1}{2}{\overline{H}}(2n,2)$ by considering the action of this automorphism group on the set $X\times X\times X$. Then, we further describe the structure of T for the case $n=2D$ and $D\ge 3$. The decomposition of T will be given by using the homogeneous components of $V{:}{=}{\mathbb {C}}^X$, each of which is a nonzero subspace of V spanned by the irreducible T-modules that are isomorphic. Moreover, we display a computable basis for every homogeneous component of V.