A Shuffle Theorem for Paths Under Any Line

TL;DR

This paper extends the shuffle theorem to paths under arbitrary lines, connecting combinatorial lattice paths with algebraic formulas involving Schiffmann algebra and LLT polynomials, generalizing previous conjectures and proofs.

Contribution

It generalizes the shuffle theorem to non-integer intercept lines and links combinatorial identities with algebraic formulas via infinite series of $GL_{l}$ characters.

Findings

Derived a new combinatorial identity as a polynomial truncation of an infinite series identity.

Connected the combinatorial paths under arbitrary lines with algebraic formulas involving Schiffmann algebra.

Established a series identity from a Cauchy identity for non-symmetric Hall-Littlewood polynomials.

Abstract

We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose $x$ and $y$ intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $GL_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for non-symmetric Hall-Littlewood polynomials.