Congruence for lattice path models with filter restrictions and long steps

TL;DR

This paper develops a path counting formula for 2D lattice models with filter restrictions and long steps, connecting combinatorics to tensor product multiplicities in quantum algebra.

Contribution

It introduces a new combinatorial approach to compute multiplicities in tensor products of quantum group modules using lattice path congruences.

Findings

Derived an explicit path counting formula for complex lattice models.

Connected combinatorial models to module multiplicities in quantum algebra.

Explored properties of region congruences in lattice path models.

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.