Ferromagnetic Ordering of Energy Levels for $U_q(\mathfrak{sl}_2)$ Symmetric Spin Chains

摘要: 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 deformation $\textrm{SU}(2)$, $q>0$. derive sufficient conditions Hamiltonian to satisfy property we call {\em Ferromagnetic Ordering Energy Levels}. This is that ground state energy restricted a fixed total subspace decreasing function spin. Using Perron-Frobenius theorem, show are positivity all interactions in dual canonical basis Lusztig. characterize cone positive showing it simplicial consisting non-positive linear combinations "cascade operators," special new $U_q(\mathfrak{sl}_2)$ intertwiners define. also applications interacting particle processes.