The Terwilliger algebra of the halved folded 2n-cube from the viewpoint of its automorphism group action

Let 1 2 H ¯ ( 2 n , 2 ) denote the halved folded 2 n -cube with vertex set X and let T : = T ( x ) denote the Terwilliger algebra of 1 2 H ¯ ( 2 n , 2 ) with respect to a fixed vertex x . In this paper, we assume n ≥ 4 and show that T coincides with the centralizer algebra of the stabilizer of x in the automorphism group of 1 2 H ¯ ( 2 n , 2 ) by considering the action of this automorphism group on the set X × X × X . Then, we further describe the structure of T for the case n = 2 D and D ≥ 3 . The decomposition of T will be given by using 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. Moreover, we display a computable basis for every homogeneous component of V .