Congruence for lattice path models with filter restrictions and long steps

Abstract

We derive a path counting formula for two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves a problem of finding an explicit formula for multiplicities of modules in tensor product decomposition of $T(1)^{\otimes N}$ for $U_q(sl_2)$ with divided powers, where $q$ is a root of unity. Combinatorial treatment of this problem calls for definition of congruence of regions in lattice path models, properties of which are explored in this paper.