**1. Introduction**

Representation theory of Kac–Moody algebras to this day serves as inspiration for numerous combinatorial problems, solutions to which give rise to interesting combinatorial structures. Examples of this can be met in [1–3] and many other well-known works. The problem of tensor power decomposition, in turn, can be considered from the combinatorial perspective as a problem of counting lattice paths in Weyl chambers [4–7]. In this paper, we count paths on the Bratteli diagram [8], reproducing the decomposition of tensor powers of the fundamental module of the quantum group *Uq*(*sl*2) with divided powers, where *q* is a root of unity ([9–12]), into indecomposable modules. Combinatorial treatment of this problem gives rise to some interesting structures on lattice path models, such as filter restrictions, first introduced in [13], and long steps, which are introduced in the present paper.

In [13], the considered lattice path model was motivated by the problem of finding explicit formulas for multiplicities of indecomposable modules in the decomposition of tensor power of fundamental module *T*(1) of the small quantum group *uq*(*sl*2) ([14]). We call this model the auxiliary lattice path model [9]. It consists of the left wall restriction at *x* = 0 and filter restrictions located periodically at *x* = *nl* − 1 for *n* ∈ N. For *n* = 1, the filter restriction is of type 1, and the rest of the values of *n* filter restrictions are of type 2. Applying periodicity conditions (*M* + 2*l*, *N*)=(*M*, *N*), *M*, *N* ≥ *l* − 1 to the Bratteli diagram of this model allows one to obtain another lattice path model, recursion for weighted numbers of paths that coincide with recursion for multiplicities of indecomposable *uq*(*sl*2)-modules in the decomposition of *T*(1)⊗*N*. Counting weighted numbers of paths descending from (0, 0) to (*M*, *N*) on this folded Bratteli diagram allows one to obtain desired formula for multiplicity, where *M* stands for the highest weight of a module, the multiplicity of which is in question, and *N* stands for the tensor power of *T*(1). This has been performed in [9].

We found that the auxiliary lattice path model can be modified in a different way, giving results for representation theory of *Uq*(*sl*2), the quantized universal enveloping

**Citation:** Solovyev, D. Congruence for Lattice Path Models with Filter Restrictions and Long Steps. *Mathematics* **2022**, *10*, 4209. https:// doi.org/10.3390/math10224209

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 5 October 2022 Accepted: 8 November 2022 Published: 11 November 2022

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algebra of *sl*<sup>2</sup> with divided powers, when *q* is a root of unity ([15]). Instead of applying periodicity conditions to the auxiliary lattice path model, as in the case of *uq*(*sl*2), for *Uq*(*sl*2) we consider all filters to be of the 1st type and also allow additional steps from *x* = *nl* − 2 to *x* = (*n* − 2)*l* − 1, where *n* ≥ 3. Counting weighted numbers of paths descending from (0, 0) to (*M*, *N*) on the Bratteli diagram of the lattice path model obtained by this modification gives a formula for the multiplicity of *T*(*M*) in the decomposition of *T*(1)⊗*N*.

The main goal of this paper is to give a more in-depth combinatorial treatment of the auxiliary lattice path model in the presence of long steps and obtain explicit formulas for weighted numbers of paths, descending from (0, 0) to (*M*, *N*). We explore combinatorial properties of long steps, as well as define boundaries and congruence of regions in lattice path models. Latter is found to be useful for deriving formulas for weighted numbers of paths. For any considered region, weighted numbers of paths at boundary points uniquely define such for the rest of the region by means of recursion. So, for congruent regions in different lattice path models, regions where, roughly speaking, recursion is similar, it is sufficient to prove identities only for boundary points of such regions.

This paper is organized as follows. In Section 2, we introduce the necessary notation. In Section 3, we give background on the auxiliary lattice path model. In Section 4, we introduce the notion of regions in lattice path models, boundary points and congruence of regions. In Section 5, we explore combinatorial properties of long steps in periodically filtered lattice path models and consider the auxiliary lattice path model in the presence of long steps. We do so by means of boundary points and congruence of regions. In Section 6, we modify the auxiliary lattice path model and argue that the recursion for the weighted number of paths in such modified model coincides with the recursion for multiplicities of modules in tensor product decomposition of *T*(1)⊗*<sup>N</sup>* for *Uq*(*sl*2) with divided powers, where *q* is a root of unity. In Section 7, we prove formulas for the weighted numbers of descending paths, relevant to this modified model. In Section 8, we conclude this paper with observations for possible future directions or research.

### **2. Notations**

In this paper, we use the notation following [16]. For our purposes of counting multiplicities in tensor power decomposition of *Uq*(*sl*2)-module *T*(1), throughout this paper, we consider the lattice

$$\mathcal{L} = \{(n, m)|n + m = 0 \bmod 2\} \subset \mathbb{Z}^2\_{\text{-}}$$

and the set of steps S = S*<sup>L</sup>* ∪ S*R*, where

$$\mathbb{S}\_{\mathbb{R}} = \{ (\mathbf{x}, \mathbf{y}) \rightarrow (\mathbf{x} + \mathbf{1}, \mathbf{y} + \mathbf{1}) \}, \\ \mathbb{S}\_{\mathbb{L}} = \{ (\mathbf{x}, \mathbf{y}) \rightarrow (\mathbf{x} - \mathbf{1}, \mathbf{y} + \mathbf{1}) \}.$$

A *lattice path* P in L is a sequence P = (*P*0, *P*1, ... , *Pm*) of points *Pi* = (*xi*, *yi*) in L with starting point *P*<sup>0</sup> and the endpoint *Pm*. The pairs *P*<sup>0</sup> → *P*1, *P*<sup>1</sup> → *P*<sup>2</sup> ... *Pm*−<sup>1</sup> → *Pm* are called steps of P.

Given starting point *A* and endpoint *B*, a set S of steps and a set of restrictions C we write

$$L(A \to B; \mathbb{S} \mid \mathcal{C})$$

for the set of all lattice paths from *A* to *B* that have steps from S and obey the restrictions from C. We denote the number of paths in this set as

$$|L(A \to B; \mathbb{S} \mid \mathcal{C})|.$$

The set of restrictions C in lattice path models considered throughout this paper mostly contain wall restrictions and filter restrictions. Left(right) wall restrictions forbid steps in the left(right) direction, reflecting descending paths and preventing them from crossing the 'wall'. Filter restrictions forbid steps in certain directions and provide other steps with non-uniform weights, so paths can cross the 'filter' in one direction, but cannot cross it in the opposite direction. A rigorous definition of these restrictions is given in subsequent sections.

To each step from (*x*, *y*) to (*x*˜, *y*˜) we assign the weight function *ω* : S −→ R><sup>0</sup> and use notation (*x*, *y*) *<sup>ω</sup>* −→ (*x*˜, *y*˜) to denote that the step from (*x*, *y*) to (*x*˜, *y*˜) has the weight *ω*. By default, all unrestricted steps from S will have weight 1 and is denoted by an arrow with no number at the top. The *weight* of a path P is defined as the product

$$
\omega(\mathcal{P}) = \prod\_{i=0}^{m-1} \omega(P\_i \to P\_{i+1}).
$$

For the set *L*(*A* → *B*; S | C) we define the *weighted number of paths* as

$$Z(L(A \to B; \mathbb{S} \mid \mathcal{C})) = \sum\_{\mathcal{P}} \omega(\mathcal{P})\_{\mathcal{H}}$$

where the sum is taken over all paths P ∈ *L*(*A* → *B*; S | C).

#### **3. The Auxiliary Lattice Path Model**

In this section, we briefly revise notions and results obtained in [13], relevant for future considerations. It is convenient for us to omit mentioning S in *L*(*A* → *B*; S | C). All paths considered below involve steps from set S unless stated otherwise.

#### *3.1. Unrestricted Paths*

Let *L*(*A* → *B*) be the set of unrestricted paths from *A* to *B* on lattice L with the steps S. An example of such a path is given in Figure 1.

**Figure 1.** Example of an unrestricted path in *L*((0, 0) → (*M*, *N*)) for lattice L and set of steps S.

**Lemma 1.** *For a set of unrestricted paths with steps* S *we have*

$$|L((0,0)\rightarrow(M,N))|=\binom{N}{\frac{N-M}{2}}.\tag{1}$$
