The Terwilliger algebra of the hypercube

Abstract

We give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube QD of dimension D. Let X denote the vertex set of QD. Fix a vertex x ∈ X, and let T = T(x) denote the associated Terwilliger algebra. We show that T is the subalgebra of Mat X(ℂ) generated by the adjacency matrix A and a diagonal matrix A* = A*(x), where A* has yy entry D - 2∂(x, y) for all y ∈ X, and where ∂ denotes the path-length distance function. We show that A and A* satisfy A2 A* - 2AA* A + A* A2 = 4A*, A*2 A - 2A* AA* + AA*2 = 4A. Using the above equations, we find the irreducible T-modules. For each irreducible T-module W, we display two orthogonal bases, which we call the standard basis and the dual standard basis. We describe the action of A and A* on each of these bases. We give the transition matrix from the standard basis to the dual standard basis for W. We compute the multiplicity with which each irreducible T-module W appears in ℂX. We give an elementary proof that QD has the Q-polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebra sl2(ℂ). We obtain an element φ of T that generates the center of T. We obtain the central primitive idempotents of Tas polynomials in φ. © 2002 Elsevier Science Ltd. All rights reserved.