Jordan cells of periodic loop models

Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, T _N, is an element of the periodic Temperley-Lieb algebra , where N is the number of sites on a section of the cylinder, and β = -q - q ^-1 = 2cos λ and the weights of contractible and non-contractible loops. The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c = 1 - 6λ ²/(π(λ - π)). The abstract element T_N acts naturally on (a sum of) spaces , similar to those upon which the standard modules of the (classical) Temperley-Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a twist parameter v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of T_N both between sectors with distinct defect numbers and within a given sector.