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

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.