Affine Transformations Of Tridiagonal Pairs Of Shape (1,3,3,1)

The concept of a tridiagonal pair originated in algebraic graph theory, or more precisely,the theory of Q-polynomial distance-regular graphs. The concept is explicit in [1] in1999.ThenBannai and Ito began a systematic study with it. Some notable papers on the topic are [2][3][4][5].Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A: V→V and A*:V*→V*that satis-fies the following conditions:（i） each of A, A*is diagonalizable;（ii） there exists an ordering{V_i}_i^d=0of the eigenspaces of A such that A*V_i∈V_i-1+V_i+V_i+1（0≤i≤d）, whereV-1=0and V_d+1=0;（iii） there exists an ordering {V_I*}_i^δ=0of the eigenspaces of A*such thatAV*_i∈V*_i-1+V*_i+V*_i+1（0≤i≤δ）, where V*_-1:=0and V*_δ+1:=0（iv） there is no subspaceW of V such that AW∈W, A*W∈W, W=0, W=V. We call such a pair a tridiagonalpair on V.Let A, A*be a tridiagonal pair. Then for p,q,p*,q*∈k with p,p*nonzero, the pair pA+qI, p*A*+q*I is also tridiagonal on V. In this thesis, I discusse the affine transformationsof the tridiagonal pair and solve part of the open problem that Tewilliger raised in his paper [9].Concretely, in the first part, we mainly introduce the concepts of tridiagonal pairs andtridiagonal systems.In the second part, we discuss the parameter arrays of the relatives of a tridiagonal systemof shape （1,3,3,1）.In the third part, we give the necessary and sufficient conditions for a tridiagonal system ofshape （1,3,3,1） to be affine isomorphic to its relatives, and list the affine isomorphism classesof it. Finally, we give the affine isomorphism classes of a tridiagonal pair of shape （1,3,3,1）.