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,Dkb(l,Dk)Kl,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.