Terwilliger Algebra Of The Johnson Scheme Revisited From Group Representations

Algebraic combinatorics is an important branch of combinatorial math-ematics.Association schemes are the core of algebraic combinatorics,which are introduced by Bose and Shimamoto,and recognized and fully used as the basic underlying structures of coding theory and design theory by Delsarte in 1973.It has been developed rapidly after the publication of the monograph"Algebraic Combinatorics I:Association Schemes" by Bannai and Ito.In the book association scheme is called "group theory without group" or "represen-tation theory of combinatorial objects".In fact,this area has been interacted with other branches,such as coding theory,design theory,graph theory,Lie algebra theory,finite groups,quantum groups.Association schemes are general structures and we believe that the most important ones are(P&Q)-polynomial schemes which are not only interesting by themselves but also important as underlying space for coding and design theory.We wish to classify(P&Q)-polynomial schemes in the future.Ter-williger made great contributions to this classification.He introduced the notion of the "subconstituent algebra" in a series of papers which is now call the Terwilliger algebra.He established the representation theory for the alge-bra in the "thin case",i.e.the theory of Leonard systems which enables us to analyze local structures of a(P&Q)-polynomial scheme algebraically in the thin case.Johnson scheme is one of the typical examples of(P&Q)-polynomial schemes that are "thin".Let J(N,D)(2D<N)be the Johnson scheme and T = T(x0)its Terwilliger algebra,where x0 is a base point.Each irreducible T-module W appears in the standard module V and gives rise to a Leonard system LS(W).The isomorphism class of LS(W)as a Leonard system determines the isomor-phism class of W as a T-module.Terwilliger claimed without proof that if d ? 1,(1)the isomorphism class of LS(W)is determined by a triple(v,?,d),where v is the endpoint,? the dual endpoint and d the diameter and(2)the triples(v,? d)of nonnegative integers satisfy 0?D-d/2?v???D-d?D,(1)d ? ?D-2v,min?D-?,N-D-2v}}(2)In this thesis,we described the triples(v,?,d)by two free parameters?,?,i.e.we established a bijection when N ? 2D between the triples(v,?,d)and the pairs(?,?)of nonnegative integers with 0???D/2,(3)0 ??min{D,N-D/2},(4)0?? + ??D.(5)We give an interpretation of the meaning of the free parameters ?,? from the viewpoint of group representations by the action of the stabilizer H of at the base point x0 in the automorphism group of J(N,D).We show that the Terwilliger's list of(v,?,d)is complete,i.e,there is a bijection between the isomorphism classes of Leonard systems LS(W)and the triples(v,?,d)allowing d = 0,i.e,a bijection between the isomorphism classes of irreducible T-modules W and the triples(v,?,d).This thesis is organized as follows.In Chapter 1,we introduce the origin of(P&Q)-polynomial schemes and the development of Terwilliger algebra.In Chapter 2,we summarize some basic knowledge of association schemes,Bose-Mesner algebras,Terwilliger algebras,P-polynomial schemes,Q-polynomial schemes and Leonard systems.In Chapter 3,we focus on a classical example of P-&Q-polynomial schemes:Johnson scheme,and give a characterization of Terwilliger algebra of J(N,D).Firstly,we discuss the action of the stabilizer H at the base point x0 in the automorphism group of J(N,D).Then we determine the structure of an irre-ducible S-module W,where S = HomH(V,V)and V is the standard module,and parameterize W as a,pair(?,?).By the two key theorems for J(N,D):(1)an irreducible S-module W is also irreducible as a T-module;(2)there is a bijection between the pairs(?,?)and the triples(v,?,d)when N ? 2D,we determine the Terwilliger algebra T of the Johnson scheme J(N,D).