The Terwilliger Algebra Of The Twisted Grassmann Scheme

In 2005,E.van Dam and J.Koolen found the twisted Grassmann scheme.It has the same parameters as the Grassmann scheme,but they are not i-somorphic.In fact,the twisted Grassmann scheme is the only example of P-polynomial association schemes of unbounded diameter known to us so far that does not allow an automorphism group which is transitive on the vertex set.In 2008,S.Bang,T.Fujisaki and J.Koolen studied the local eigenvalues of the twisted Grassmann scheme on the first subconstituent and they showed that the twisted Grassmann scheme is a counterexample to Terwilliger's con-jecture:The Terwilliger algebra of P-polynomial association scheme does not depend on the choice of the base vertex.In fact,the automorphism group of the twisted Grassmann scheme has two orbits on the vertex set and the Terwilliger algebra depends on which orbit the base vertex belongs to.In this thesis,we determine the structure of the irreducible modules of endpoint 1 for the Terwilliger algebra of the Grassmann scheme and twisted Grassmann scheme.For twisted Grassmann scheme,for a base vertex in one of the orbits,The Terwilliger algebra has four non-isomorphic irreducible modules of endpoint 1 and all of them are thin.For a base vertex in the other orbit,the Terwilliger algebra has six non-isomorphic irreducible modules of endpoint 1:Three of them are thin and the rest are non-thin.We analyse the structure of the T-modules from the viewpoint of Leonard pairs or tridiagonal pairs.We remark that paper[1]made a mistake about the multiplicities of the eigenvalues on the first subconstituent,which we corrected,this correction has opened a new research,area concerning the graph isomorphism and the standard T-module(see the closing remark of Chapter 5 for the details).This thesis is organized as follows:In chapter 1,we explain where(P and Q)-polynomial association schemes come from,and some background of twisted Grassmann graph;in chapter 2,we summarize basic knowledge of association schemes,Bose-Mesner algebras,Terwilliger algebras,(P and Q)-polynomial association schemes,Leonard systems and distance-regular graph;in chapter 3,we summarize the Terwilliger theory about T-modules of endpoint 1 for a(P and Q)-polynomial association scheme with proofs;in chapter 4,we give the structure of thin irreducible T-modules with endpoint 1 for a distance-regular graph in terms of the local eigenvalue;in chapter 5,we study the structure of irreducible modules of Grassmann scheme J_q(2e + 1,e)and twisted Grassmann scheme(?)_q(2e + 1,e);in chapter 6,1 summarize what I did and make a new goal for the future.