The terwilliger algebra of a distance-regular graph - Go, Junie T. (Animo Repository, 1999)
This dissertation deals with the Terwilliger algebra of a distance-regular graph.The study has two main parts. The first part studies the Terwilliger algebra of the D-cube QD, also known as hypercube. Let X denote the vertex set of QD. Fix x e X, and let T=T(x) denote its associated Terwilliger algebra. T is shown as the subalgebra of Matx (C) generated by the adjacency matrix A and a diagonal matrix A*=A*(x), where A* has yy entry D-2a(x,y) for all y e X. A, A* satisfyA2A*-2AA*+A*A2 = 4A*,A*2-2A*AA*+AA*2 = 4AUsing the above equations, the irreducible T-modules is found. For each irreducible T-module W, two orthogonal bases are displayed, the standard basis and the dual standard basis. Action of A and A* are described on these basis. The transition matrix is given from the standard basis to the dual standard basis. The multiplicity with which each irreducible T-module W appears in is computed. An elementary proof that QD has the Q-polynomial property is given. T, a homomorphic image of the universal enveloping algebra of the Lie algebra sl2 (C) is shown. The center of T is described. The second part of this dissertation studies the Terwilliger algebra of a tight distance-regular graph. Let r = (X,R) denote a distance-regular graph with diameter D greater than or equal to 3. Fix x e C, and let T - T(x) denote its associated Terwilliger algebra. We associate two integer parameters: the endpoint and the diameter, to each irreducible T-module. It turns out that the dimension of such a module is at least one more than its diameter. Whenever equality is attained, the module is said to be thin. To each irreducible T-module of endpoint 1 and diameter D-2, another real parameter, the type is associated. The assumption now is that r is tight. The r is shown to have at least one irreducible T-module of type 1, at least one irreducible T-module of type D, and up to isomorphism, no other irreducible T-modules of endpoint 1. Each type is shown to be thin and has diameter D-2. The multiplicity with which each module appears inCx is computed.