Dens, nests and the Loehr-Warrington conjecture

TL;DR

This paper develops a combinatorial formula involving nested lattice paths and LLT polynomials, proving and generalizing the Loehr-Warrington conjecture related to symmetric functions and the elliptic Hall algebra.

Contribution

It introduces a new nests-in-a-den formula that generalizes previous shuffle theorems and proves the Loehr-Warrington conjecture for arbitrary parameters.

Findings

Proves the Loehr-Warrington conjecture for the $(m,n)$ case.

Unifies previous shuffle theorems into a single generalized formula.

Establishes a combinatorial framework using nests and LLT polynomials.

Abstract

In a companion paper, we introduced raising operator series called Catalanimals. Among them are Schur Catalanimals, which represent Schur functions inside copies $\Lambda (X^{m,n})\subset \mathcal{E} $ of the algebra of symmetric functions embedded in the elliptic Hall algebra $\mathcal{E} $ of Burban and Schiffmann. Here we obtain a combinatorial formula for symmetric functions given by a class of Catalanimals that includes the Schur Catalanimals. Our formula is expressed as a weighted sum of LLT polynomials, with terms indexed by configurations of nested lattice paths called nests, having endpoints and bounding constraints controlled by data called a den. Applied to Schur Catalanimals for the alphabets $X^{m,1}$ with $n=1$, our `nests in a den' formula proves the combinatorial formula conjectured by Loehr and Warrington for $\nabla^m s_{\mu }$ as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. When $n$ is arbitrary, our formula establishes an $(m,n)$ version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduces to our previous shuffle theorem for paths under any line. Both this and the $(m,n)$ Loehr-Warrington formula generalize the $(km,kn)$ shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our formula here unifies these two generalizations.