Tridiagonal pairs of type III with height one

Abstract

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A^∗ denote a tridiagonal pair on V  of diameter d.  Let V₀, . . . , V_d  denote a standard ordering of  the eigenspaces of A on V , and let θ₀, . . . , θ_ddenote the corresponding eigenvalues of A. It is assumed that d ≥ 3.  Let ρ_i  denote the dimension of V_i. The sequence ρ₀, ρ₁, . . . , ρ_dis called the shape of the tridiagonal pair. It is known that ρ₀ = 1 and there  exists  a  unique  integer  h (0 ≤ h ≤ d/2)  such  that  ρ_i−1< ρ_i  for  1 ≤ i ≤ h,  ρ_i−1 = ρ_i  for  h < i ≤ d − h,  and  ρ_i−1> ρ_ifor d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1.  In each case, an interesting basis is found for V and the actions of A, A^∗ on this basis are described.