  We consider the class of quantum spin chains with arbitrary
$U_q(\mathfrak{sl}_2)$-invariant nearest neighbor interactions, sometimes
called $\textrm{SU}_q(2)$ for the quantum deformation of $\textrm{SU}(2)$, for
$q>0$. We derive sufficient conditions for the Hamiltonian to satisfy the
property we call {\em 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
$U_q(\mathfrak{sl}_2)$ intertwiners we define. We also state applications to
interacting particle processes.
