Equating Inv-Quinv Formulas for the $q$-Whittaker and Modified Hall-Littlewood Functions

Abstract

The two combinatorial formulas for modified Macdonald polynomials given by Haglund, Haiman, and Loehr (2005) and by Ayyer, Mandelshtam, and Martin (2023) give two combinatorial interpretations of $q$-Whittaker functions and modified Hall-Littlewood functions. The main result of this paper is a combinatorial proof of the equality between these two formulas, using weighted Dyck path symmetric functions (introduced by Carlsson and Mellit, 2018) as intermediate objects. In the final section, we remark on the Schur positivity of these weighted Dyck path symmetric functions.