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, . . . , Vd denote a standard ordering of the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3. Let ρi denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is 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 ρi−1 < ρi for 1 ≤ i ≤ h, ρi−1 = ρi for h < i ≤ d − h, and ρi−1 > ρi for 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.