Tridiagonal pairs of height one

AbstractLet (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