Two conjectures for Macdonald polynomials: The stretching symmetry and Haglund's conjecture

Seung Jin Lee, Jaeseong Oh, Brendon Rhoades

Published 2022-03-09Version 1

We propose a conjecture which is a symmetry relation for the modified Macdonald polynomials of stretched partitions. We prove the conjecture for one-column or one-row shape in two ways: using the LLT expansion of the modified Macdonald polynomials and using the Garsia--Haiman module. In addition, we prove a conjecture of Haglund concerning multi-$t$-Macdonald polynomials of two rows.