Counting filter restricted paths in $\mathbb{Z}^2$ lattice

TL;DR

This paper develops an explicit formula for counting weighted paths in a 2D lattice with filter restrictions, motivated by quantum algebra representation problems, advancing combinatorial and algebraic understanding.

Contribution

It introduces a novel path counting formula for lattice models with filter restrictions, connecting combinatorics with quantum algebra representation theory.

Findings

Derived an explicit path counting formula with weights

Connected lattice path enumeration to tensor power decomposition

Motivated by quantum $ ext{sl}_2$ at roots of unity

Abstract

We derive a path counting formula for two-dimensional lattice path model on a plane with filter restrictions. A filter is a line that restricts the path passing it to one of possible directions. Moreover, each path that touches this line is assigned a special weight. The periodic filter restrictions are motivated by the problem of tensor power decomposition for representations of quantum $\mathfrak{sl}_2$ at roots of unity. Our main result is the explicit formula for the weighted number of paths from the origin to a fixed point between two filters in this model.