Equating Inv-Quinv formulas for the $q$-Whittaker and modified Hall-Littlewood functions

TL;DR

This paper demonstrates the equivalence of two different formula sets for $q$-Whittaker and modified Hall-Littlewood functions using weighted path symmetric functions, unifying prior combinatorial approaches.

Contribution

It establishes the equality between Inv and Quinv formulas for these functions through the application of weighted path symmetric functions.

Findings

Proves the equivalence of Inv and Quinv formulas

Unifies combinatorial formulas for $q$-Whittaker and Hall-Littlewood functions

Utilizes weighted path symmetric functions for the proof

Abstract

We explain the equality between the two sets of formulas for $q$-Whittaker functions and modified Hall-Littlewood functions obtained by Haglund, Haiman and Loehr - the Inv formula and Ayyer, Mandelshtam and Martin - the Quinv formula by use of weighted path symmetric functions introduced by Carlsson and Mellit.