Spin Leonard pairs

Abstract

Let ${\mathbb K}$ denote a field, and let V denote a vector space over ${\mathbb K}$ of finite positive dimension. A pair A, A* of linear operators on V is said to be a Leonard pair on V whenever for each B∈{A, A*}, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A* on V is said to be a spin Leonard pair whenever there exist invertible linear operators U, U* on V such that UA = A U, U*A* = A*U*, and UA* U ⁻¹ = U*⁻¹ AU*. In this case, we refer to U, U* as a Boltzmann pair for A, A*. We characterize the spin Leonard pairs. This characterization involves explicit formulas for the entries of the matrices that represent A and A* with respect to a particular basis. The formulas are expressed in terms of four algebraically independent parameters. We describe all Boltzmann pairs for a spin Leonard pair in terms of these parameters. We then describe all spin Leonard pairs associated with a given Boltzmann pair. We also describe the relationship between spin Leonard pairs and modular Leonard triples. We note a modular group action on each isomorphism class of spin Leonard pairs.