  We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ
spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop
algebra symmetry if the $q$ parameter is given by a root of unity,
$q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate
eigenspace generated by a regular Bethe state in some sectors, rigorously as
follows: We show that every regular Bethe ansatz eigenvector in the sectors is
a highest weight vector and derive the highest weight ${\bar d}_k^{\pm}$, which
leads to evaluation parameters $a_j$. If the evaluation parameters are
distinct, we obtain the dimensions of the highest weight representation
generated by the regular Bethe state.
