Character decomposition of Potts model partition functions, I: Cyclic geometry

We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L×NL×N, with boundary conditions that are free in the L-direction and periodic in the N  -direction. The decomposition of the partition function in terms of the characters K1+2lK1+2l (with l=0,1,…,Ll=0,1,…,L) has previously been studied using various approaches (quantum groups, combinatorics, transfer matrices). We first show that the K1+2lK1+2l thus defined actually coincide, and can be written as traces of suitable transfer matrices in the cluster picture. We then proceed to similarly decompose constrained partition functions in which exactly j clusters are non-contractible with respect to the periodic lattice direction, and a partition function with fixed transverse boundary conditions.