  The spectra of recently constructed auxiliary matrices for the six-vertex
model respectively the spin s=1/2 Heisenberg chain at roots of unity q^N=1 are
investigated. Two conjectures are formulated both of which are proven for N=3
and are verified numerically for several examples with N>3. The first
conjecture identifies an abelian subset of auxiliary matrices whose eigenvalues
are polynomials in the spectral variable. The zeroes of these polynomials are
shown to fall into two sets. One consists of the solutions to the Bethe ansatz
equations which determine the eigenvalues of the six-vertex transfer matrix.
The other set of zeroes contains the complete strings which encode the
information on the degeneracies of the model due to the loop symmetry sl_2
present at roots of 1. The second conjecture then states a polynomial identity
which relates the complete string centres to the Bethe roots allowing one to
determine the dimension of the degenerate eigenspaces. Its proof for N=3
involves the derivation of a new functional equation for the auxiliary matrices
and the six-vertex transfer matrix. Moreover, it is demonstrated in several
explicit examples that the complete strings coincide with the classical
analogue of the Drinfeld polynomial. The latter is used to classify the
finite-dimensional irreducible representations of the loop algebra sl_2. This
suggests that the constructed auxiliary matrices not only enable one to solve
the six-vertex model but also completely characterize the decomposition of its
eigenspaces into irreducible representations of the underlying loop symmetry.
