Tridiagonal pairs of height one

Let (A, A*) denote a tridiagonal pair on a vector space V over a field K. Let V0, … , Vd denote a standard ordering of the eigenspaces of A on V, and let θ0, … , θd denote the corresponding eigenvalues of A. We assume d ⩾ 3. Let q denote a scalar taken from the algebraic closure of K such that q2 + q−2 + 1 = (θ3 − θ0)/(θ2 − θ1). We assume q is not a root of unity. Let ρi denote the dimension of Vi. The sequence ρ0, ρ1, … , ρd is called the shape of the tridiagonal pair. It is known 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 we show that the shape of a tridiagonal pair of height one with ρ0 = 1 is either 1, 2, 2, … , 2, 1 or 1, 3, 3, 1. In each case, we display a basis for V and give the action of A, A* on this basis.