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We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain offinite length N. For the case that q is a root of unity, we propose a formulafor the number of admissible solutions of the Bethe ansatz equations in termsof dimensions of irreducible representations of the Temperley-Lieb algebra; anda formula for the degeneracies of the transfer matrix eigenvalues in terms ofdimensions of tilting sl(2)_q-modules. These formulas include corrections thatappear if two or more tilting modules are spectrum-degenerate. For the XX case(q=exp(i pi/2)), we give explicit formulas for the number of admissiblesolutions and degeneracies. We also consider the cases of generic q and theisotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8are presented. Our results are consistent with the Bethe ansatz solution beingcomplete.

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