International Journal of Modern Physics B, ❢c World Scientific Publishing Company

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2Nη = m1 + im2τ , where N,m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 (mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 (mod N) of the six-vertex model at Nth roots of unity is given by 22S Z max/N , where SZ max is the maximal value of the total spin operator SZ in the degenerate eigenspace.