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The spectra of recently constructed auxiliary matrices for the six-vertexmodel respectively the spin s=1/2 Heisenberg chain at roots of unity q^N=1 areinvestigated. Two conjectures are formulated both of which are proven for N=3and are verified numerically for several examples with N>3. The firstconjecture identifies an abelian subset of auxiliary matrices whose eigenvaluesare polynomials in the spectral variable. The zeroes of these polynomials areshown to fall into two sets. One consists of the solutions to the Bethe ansatzequations which determine the eigenvalues of the six-vertex transfer matrix.The other set of zeroes contains the complete strings which encode theinformation on the degeneracies of the model due to the loop symmetry sl_2present at roots of 1. The second conjecture then states a polynomial identitywhich relates the complete string centres to the Bethe roots allowing one todetermine the dimension of the degenerate eigenspaces. Its proof for N=3involves the derivation of a new functional equation for the auxiliary matricesand the six-vertex transfer matrix. Moreover, it is demonstrated in severalexplicit examples that the complete strings coincide with the classicalanalogue of the Drinfeld polynomial. The latter is used to classify thefinite-dimensional irreducible representations of the loop algebra sl_2. Thissuggests that the constructed auxiliary matrices not only enable one to solvethe six-vertex model but also completely characterize the decomposition of itseigenspaces into irreducible representations of the underlying loop symmetry.Comment: 24 pages, continuation of math-ph/030200

Auxiliary matrices for the six-vertex model atqN= 1: II. Bethe roots, complete strings and the Drinfeld polynomial

Auxiliary matrices for the six-vertex model atq^N= 1: II. Bethe roots, complete strings and the Drinfeld polynomial