The Terwilliger algebra of the hypercube

  • Authors:

  • Junie T. Go

  • Affiliations:

  • College of Education, University of St La Salle, La Salle Avenue, Bacolod City, Negros Occidental, Philippines

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2002

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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 MatX (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 A2A* - 2AA*A +A*A2 = 4A*, A*2A - 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 mulztiplicity 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 T as polynomials in φ