Eigenvalue amplitudes of the Potts model on a torus

We consider the Q-state Potts model in the random-cluster formulation, defined on finite two-dimensional lattices of size L × N with toroidal boundary conditions. Due to the non-locality of the clusters, the partition function Z(L,N) cannot be written simply as a trace of the transfer matrix TL. Using a combinatorial method, we establish the decomposition Z(L,N) = ∑ l,Dk bKl,Dk , where the characters Kl,Dk = ∑ i(λi) N are simple traces. In this decomposition, the amplitudes b(l,Dk) of the eigenvalues λi of TL are labelled by the number l = 0, 1, . . . , L of clusters which are non-contractible with respect to the transfer (N) direction, and a representation Dk of the cyclic group Cl. We obtain rigorously a general expression for b (l,Dk) in terms of the characters of Cl, and, using number theoretic results, show that it coincides with an expression previously obtained in the continuum limit by Read and Saleur.