Affine Transformations Of A Sharp Tridiagonal Pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. By a tridiagonal pair, we mean an ordered pair A, A* of K-linear trans-formations from V to V that satisfy the following conditions:(ⅰ) each of A, A* is di-agonalizable; (ⅱ) there exists an ordering {Vi}i=0d of the eigenspaces of A such that A*Vi(?)Vi-i+Vi+Vi+1 (0≤i≤d), where V-1= 0, Vd+1= 0; (ⅲ) there exists an ordering {Vi*}i=0δ of the eigenspaces of A* such that AVi*(?)Vi-1*+Vi*+Vi+1* (0≤ i≤δ), where V-1*= 0, Vδ+1*= 0; (ⅳ) there is no subspace W of V such that AW(?)W, A*W(?) W, W t≠0, W≠V. It is known that ηA+μI, η*A*+μ*I is also a tridiagonal pair on V, where η,μ,η*,μ* are scalars in K with η,η* nonzero.In this paper, we study the affine transformations of a sharp tridiagonal pair and we give the necessary and sufficient conditions for these tridiagonal pairs to be isomor-phic to A, A* or A*, A. Then we solve the open problem proposed by P. Terwilliger in [24].