Path operators and $(q,t)$-tau functions

TL;DR

This paper introduces a new class of symmetric function operators linked to lattice paths, which generalize hypergeometric tau functions and provide novel proofs for existing conjectures in algebraic combinatorics.

Contribution

The authors construct path operators with two deformation parameters, connect them to shuffle algebra elements, and apply them to deformations of hypergeometric tau functions, advancing the understanding of symmetric functions.

Findings

Path operators are associated with lattice paths and act on symmetric functions.

The $(q,t)$-deformed hypergeometric tau functions are characterized by differential equations.

A new proof of the extended delta conjecture is provided using these operators.

Abstract

We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up and down. We demonstrate that positive linear combinations of these operators are the images of Negut elements via a representation of the shuffle algebra acting on the space of symmetric functions. Additionally, we provide a monomial, elementary, and Schur symmetric function expansion for the symmetric function obtained through repeated applications of the path operators on $1$. We apply path operators to investigate a $(q,t)$-deformation of the classical hypergeometric tau functions, which generalizes several important series already present in enumerative geometry, gauge theory, and integrability. We prove that this function is uniquely characterized by a family of partial differential equations derived from a positive linear combination of path operators. We also use our operators to offer a new, independent proof of the key result in establishing the extended delta conjecture of Haglund, Remmel, and Wilson.