Bijections between different combinatorial models for $q$-Whittaker and modified Hall-Littlewood polynomials

Abstract

We consider the monomial expansion of the $q$-Whittaker and modified Hall-Littlewood polynomialsarising from specialization of the modified Macdonald polynomial. The two combinatorial formulas for the latter due to Haglund, Haiman, and Loehr and Ayyer, Mandelshtam and Martin give rise to two different parameterizing sets in each case. We produce bijections between the parameterizing sets which preserve the content and major index statistics. We identify the major index with the charge or cocharge of appropriate words and use descriptions of the latter due to Lascoux-Sch$\ddot{\text{u}}$tzenberger and Killpatrick to show that our bijections have the desired properties.