  We uncover a connection between two seemingly separate subjects in integrable
models: the representation theory of the affine Temperley-Lieb algebra, and the
algebraic structure of solutions to the Bethe equations of the XXZ spin chain.
We study the solution of Bethe equations analytically by computational
algebraic geometry, and find that the solution space encodes rich information
about the representation theory of Temperley-Lieb algebra. Using these
connections, we compute the partition function of the completely-packed loop
model and of the closely related random-cluster Potts model, on medium-size
lattices with toroidal boundary conditions, by two quite different methods. We
consider the partial thermodynamic limit of infinitely long tori and analyze
the corresponding condensation curves of the zeros of the partition functions.
Two components of these curves are obtained analytically in the full
thermodynamic limit.
