Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

We consider the U_qsl (2)-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley.Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting U_qsl (2)-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q = e^iπ/2), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q → 1) limit. Numerical solutions of the Bethe equations up to N = 8 are presented. Our results are consistent with the Bethe ansatz solution being complete.