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

Let 1/2H(n, 2) denote the halved n-cube with vertex set X and let T := T(x(0)) denote the Terwilliger algebra of 1/2H(n, 2) with respect to a fixed vertex x(0) is an element of X. In this paper, we assume n >= 6. We first characterize T by considering the action of the automorphism group of 1/2H(n, 2) on the set X x X x X. We show that T coincides with the centralizer algebra of the stabilizer of x(0) in the automorphism group, and display three subalgebras of T further. Then we study the homogeneous components of V := C-X, each of which is a nonzero subspace of V spanned by the irreducible T-modules that are isomorphic. We give a computable basis for any homogeneous component of V. Finally, we describe the decomposition of T via its block-diagonalization and give a basis for the center of T by using the above homogeneous components of V. (c) 2021 Elsevier Ltd. All rights reserved.