The Terwilliger algebra of the halved cube

Abstract

Let $D\geq 3$ 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 ${\rm Mat}_X(\mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) \qquad \hbox{for all $y\in X$}. $$ In this paper we decompose the standard $\mathcal T$-module into a direct sum of irreducible $\mathcal T$-modules.