The Terwilliger Algebra of the Hypercube

We give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube QDof dimension D. Let X denote the vertex set ofQD . Fix a vertex x ∈ X, and letT = T(x) denote the associated Terwilliger algebra. We show thatT is the subalgebra of MatX(C) generated by the adjacency matrixA and a diagonal matrix A * = A * (x), where A * has yy entryD − 2 ∂(x, y) for all y ∈ X , and where ∂ denotes the path-length distance function. We show that A andA * satisfy A2A * − 2AA * A + A * A2& = & 4A * , A * 2A − 2A * AA * + AA * 2& = & 4 A. 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 andA * 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 inCX . We give an elementary proof that QDhas the Q -polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebrasl2 (C). We obtain an element φ of T that generates the center ofT . We obtain the central primitive idempotents of T as polynomials in φ.