The Terwilliger algebra of the halved cube

Abstract

Let $D\ge 3$ denote an integer. For any $x\in {\mathbb{F}}_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of ${\mathbb{F}}_2^D$ consisting of all $x\in {\mathbb{F}}_2^D$ with even $w(x)$. The $D$-dimensional halved cube $\frac{1}{2}H(D,2)$ is a finite simple connected graph with vertex set $X$ and $x,y\in X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $x\in X$. The Terwilliger algebra $\mathcal{T}=\mathcal{T}(x)$ of $\frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${\mathrm{Mat}}_X(\mathbb{C})$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^{\ast }=A^{\ast }(x)$ where $A^{\ast }$ is a diagonal matrix with $$A_{yy}^{\ast }=D-2w(x-y)for\; ally\in X.$$ In this paper we decompose the standard $\mathcal{T}$-module into a direct sum of irreducible $\mathcal{T}$-modules.