The Jordan structure of two dimensional loop models

We show how to use the link representation of the transfer matrix D_N of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter β=2cos(π(1−a/b)),a,b∈ℕ and, more specifically, partition functions of the corresponding Q-Potts spin models, with Q=β^2. The braid limit of D_N is shown to be a central element F_N(β) of the Temperley-Lieb algebra TL_N(β), its eigenvalues are determined and, for generic β, a basis of its eigenvectors is constructed using the Wenzl-Jones projector. To any element of this basis is associated a number of defects d, 0≤d≤N, and the basis vectors with the same d span a sector. Because components of these eigenvectors are singular when b∈ℤ∗ and a∈2ℤ+1, the link representations of F_N and D_N are shown to have Jordan blocks between sectors d and d′ when d−d′<2b and (d+d′)/2≡b−1 mod 2b (d>d′). When a and b do not satisfy the previous constraint, D_N is diagonalizable.