Some Research On Representations Of The Terwilliger Algebras

Tridiagonal pairs were introduced by T. Ito, K. Tanabe and P. Terwilliger to deal with the problem about the classification of distance regular graphs. Also, The notation of Tridiago-nal pairs is a mild generalization of Leonard pairs. P. Terwilliger and T. Ito give the relative notions about inverting pair, Hessenberg pair, cycle Leonard pair and Leonard triple during their study of Tridiagonal pairs. These work is very important in the classification of distance regular graphs. And they are closely related to Lie algebras, quantum groups and their rep-resentation theory. The theory is called representations of the Terwilliger algebra by E.R van Dam, J.H Koolen, H. Tanaka in the paper of Distance-regular Graphs.This paper mainly discuss the trace of the product of primitive idempotents in sharp Tridiagonal system, the classification of the q-inverting pairs of shape(1,2,1), and the con-struction of Leonard triples of Krawtchouk type. The paper is composed of five chapters and organized as follows:In Chapter1, we introduce the background, and describe others’works on representa-tions of the Terwilliger algebras. Then we list the problems that will be considered and the main results we got.In Chapter2, we give the formulae of the trace of the product of primitive idempotents in sharp Tridiagonal system by using the parameter array. We solve the problem of the trace of the product of primitive idempotents in sharp Tridiagonal system proposed by K. Nomura and P. Terwilliger.In Chapter3, we define a pair of scalars called the parameter pair for a g-inverting pair K, K*of shape(1,2,1). Then we classify the q-inverting pairs of shape(1,2,1) in terms of the parameter pairs.In Chapter4, based on the Leonard pair of Krawtchouk type A, A*, we construct Leonard triples of Krawtchouk type A, A*, AεIn Chapter5, we list some problems which will be further studied.