Tridiagonal pairs of height one

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Let ( A, A*) denote a tridiagonal pair on a vector space V over a field K . Let V 0, … , V d denote a standard ordering of the eigenspaces of A on V, and let θ 0, … , θ d denote the corresponding eigenvalues of A. We assume d ⩾ 3. Let q denote a scalar taken from the algebraic closure of K such that q 2 + q −2 + 1 = ( θ 3 − θ 0)/( θ 2 − θ 1). We assume q is not a root of unity. Let ρ i denote the dimension of V i . The sequence ρ 0, ρ 1, … , ρ d is called the shape of the tridiagonal pair. It is known 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 i 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 ρ 0 = 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.