The Terwilliger Algebra Of The Johnson Scheme

Algebraic Combinatorics is a relatively young area that emerged in the present form in the 1980’s with the publication of a book "Algebraic Combina-torics Ⅰ:Association scheme"by Eiichi Bannai and Tatsuro Ito.The theoretical origins of this subject lie in group representation theory,and the original ap-plications were to codes and designs.Lately,this area has been developing rapidly,interacting with many other branches of mathematics,such as Lie algebra theory,quantum groups and statistical mechanics.The book of Bannai and Ito proposed and discussed in detail the classi-fication problem of(P and Q)-polynomial schemes.The classification is now the central problem in algebraic combinatorics:(P and Q)-polynomial schemes are not only interesting for their own sake but also important as underlying space for coding/design theory.Most of the important contributions to this problem were made by Terwilliger.In particular,he introduced the notion of the subconstituent algebra,which we now call the Terwilliger algebra or simply the T-algebra,and he established the representation theory for it in the "thin case".This is the theory of Leonard systems[23,24,25].The Terwilliger algebra contains the Bose-Mesner algebra.In fact,it is much bigger than the Bose-Mesner algebra and contains much more information about combinatori-al local structures;the theory of Leonard systems enables us to analyze local structures of a(P and Q)-polynomial scheme in detail algebraically,when the scheme is "thin".In[25],Terwilliger determined what sort of Leonard systems appear in the known examples of(P and Q)-polynomial schemes that are "thin".The John-son scheme J(N,D)(2D<N)is one of such examples.Let(?)=(X,{Ri}0≤i≤D)be the Johnson scheme J(N,D)(2D≤N)[2,Chapter 3,Section 2]:we are given a set Ω(|Ω|=N),andFor a fixed base point x0 ∈X,let T = T(x0)be the Terwilliger algebra of X.Let V=Cx be the standard module for T.Since T.is a semi-simple algebra and V is a fait,hful T-module,all the irreducible T-modules appear in V as T-submodules up to isomorphism.Let W be an irreducible T-submodule of V.Then a Leonard system LS(W)arises from W in such a way that the isomorphism class of the irreducible T-module W is determined by the isomorphism class of the Leonard system LS(W),and vice versa[24].The isomorphism class of the Leonard system LS(W)is determined in[25,Example 6.1(1),page 198]by an ordered triple,(v,μ,d)of non-negative integers v,μ,d such thatIn this thesis,we introduce an ordered pair(α,β)of non-negative integersα,β such that and we establish a bijection between the triples(v,μ,d)and the pairs(α,β).The triple(v,μ,d)has a natural meaning from the viewpoint of Leonard systems.The pair(α,β)is expected to have a natural meaning from the view-point of representations of symmetric groups.The bijection between the triples(v,μ,d)and the pairs(α,β)is expected to be the bridge that connects group representation theory to the Terwillger algebra,and have to combinatorics.This thesis is organized as follows.In chapter 1,we explain where(P and Q)-polynomial schemes come from and why they are important.In chapter 2,we summarize basics of association schemes,Bose-Mesner algebras,Terwilliger algebras,P/Q-polynomial schemes,Leonard systems.In chapter 3,we discuss irreducible modules for the Terwilliger algebra of the Johnson scheme J(N,D)and give the bijection between the triples(α,β)and the pairs(v,μ,d).