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 beta = 2cos(pi(1 - a/b)), a, b is an element of N and, more specifically, partition functions of the corresponding Q-Potts spin models, with Q = beta(2). The braid limit of D-N is shown to be a central element F-N(beta) of the Temperley-Lieb algebra TLN(beta), its eigenvalues are determined and, for generic beta, a basis of its eigenvectors is constructed using the Wenzl-Jones projector. With 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 is an element of Z* and a is an element of 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 equivalent to b - 1 mod2b (d > d'). When a and b do not satisfy the previous constraint, D-N is diagonalizable.