LLT polynomials in the Schiffmann algebra

TL;DR

This paper connects combinatorially defined rational functions to LLT polynomials within the Schiffmann algebra, providing explicit formulas for the nabla operator on LLT polynomials, advancing understanding in symmetric functions and algebraic combinatorics.

Contribution

It establishes a new isomorphism linking rational functions to LLT polynomials in the elliptic Hall algebra, and derives explicit raising operator formulas for the nabla operator on LLT polynomials.

Findings

Explicit raising operator formula for nabla on LLT polynomials

Connection between rational functions and LLT polynomials in Schiffmann algebra

Foundation for proving Loehr-Warrington conjecture

Abstract

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished 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. As a corollary, we deduce an explicit raising operator formula for the $\nabla$ operator applied to any LLT polynomial. In particular, we obtain a formula for $\nabla ^m s_\lambda$ which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one.