Ferromagnetic Ordering of Energy Levels for Uq(sl2) Symmetric Spin Chains

We consider the class of quantum spin chains with arbitrary Uq(sl2)-invariant nearest neighbor interactions, sometimes called SUq(2) for the quantum deformation of SU(2), for q > 0. We derive sufficient conditions for the Hamiltonian to satisfy the property we call Ferromagnetic Ordering of Energy Levels. This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron-Frobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of “cascade operators,” a special new basis of Uq(sl2) intertwiners we define.