The Terwilliger algebra of the halved cube

Let $Dgeq 3$ denote an integer. For any $xin mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $mathbb F_2^D$ consisting of all $xin mathbb F_2^D$ 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 $x,yin X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $xin X$. The Terwilliger algebra $mathcal T=mathcal T(x)$ of $frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${rm Mat}_X(mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) qquad hbox{for all $yin X$}. $$ In this paper we decompose the standard $mathcal T$-module into a direct sum of irreducible $mathcal T$-modules.