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

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.

Figures (13)

• Figure 1: Bratteli diagram and a lattice path.
• Figure 2: The arrangement of restrictions on $\mathcal{L}$: the wall $\mathcal{W}_0^L$, the filters $\mathcal{F}_{l-1}^1$ and the filters $\mathcal{F}_{nl-1}^2$, where $n=2,3,\dots$ and $l=5$. Red arrows correspond to steps with weight 2.
• Figure 3: Path in $L((0,0)\to (M,N))$.
• Figure 4: Counting paths with the wall restriction. The initial path $\mathcal{P}$(red) and partially reflected path $\phi \mathcal{P}$(green).
• Figure 5: Filter $\mathcal{F}_d^{1}$. Red arrows correspond to step $(d+1,y+1)\xrightarrow[]{2}(d,y+2)$ that has a weight $2$. Other steps have weight $1$.

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