Geometric algebra and algebraic geometry of loop and Potts models

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