Schur positivity of the nabla operator on two-column modified Hall--Littlewood polynomials

TL;DR

This paper proves Schur positivity of the nabla operator acting on two-column modified Hall--Littlewood polynomials, confirming conjectures and extending results to all positive powers of nabla.

Contribution

It resolves two conjectures on Schur positivity for two-column cases and extends the results to all powers of the nabla operator.

Findings

Confirmed Schur positivity for two-column modified Hall--Littlewood polynomials

Extended positivity results to arbitrary powers of the nabla operator

Resolved two key conjectures in algebraic combinatorics

Abstract

In this paper, we investigate the Schur positivity of modified Hall--Littlewood polynomials indexed by two-column partitions under the action of the $\nabla$ operator. Specifically, we resolve two conjectures posed by Bergeron, Garsia, Haiman, and Tesler in the two-column case. Furthermore, our approach demonstrates that these results can be extended to arbitrary powers $\nabla^k$ for all integers $k\geq 1$.