The Terwilliger algebra of the halved cube

Abstract

Let D ≥ 3 denote an integer. For any x ∈ F D 2 let w(x) denote the Hamming weight of x. Let X denote the subspace of F D 2 consisting of all x ∈ F D 2 with even w(x). The D-dimensional halved cube 1 2 H(D, 2) is a finite simple connected graph with vertex set X and x, y ∈ X are adjacent if and only if w(x − y) = 2. Fix a vertex x ∈ X. The Terwilliger algebra T = T (x) of 1 2 H(D, 2) with respect to x is the subalgebra of Mat X (C) generated by the adjacency matrix A and the dual adjacency matrix A * = A * (x) where A * is a diagonal matrix withA * yy = D − 2w(x − y) for all y ∈ X. In this paper we decompose the standard T -module into a direct sum of irreducible Tmodules.