Affine transformations of a sharp tridiagonal pair

Abstract

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 transformations from V to V that satisfy the following conditions: (i) each of A,A⁎ is diagonalizable; (ii) there exists an ordering {Vi}i=0d of the eigenspaces of A such that A⁎Vi⊆Vi−1+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≠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 give the necessary and sufficient conditions for these tridiagonal pairs to be isomorphic to A,A⁎ or A⁎,A. We do this under a mild assumption, called the sharp condition.