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Schur positivity of the nabla operator on two-column modified Hall--Littlewood polynomials
Abstract
Bergeron, Garsia, Haiman, and Tesler (1999) introduced two Schur positivity conjectures concerning the action of the ∇ operator on modified Hall--Littlewood polynomials. Although the proofs of the celebrated compositional shuffle conjecture and its rational generalization, along with various symmetric function identities, have largely resolved these conjectures in the affirmative, we provide an elegant alternative proof driven by q-series techniques. Through this approach, we derive explicit Schur function expansions for two-column partitions and extend our results to arbitrary positive powers ∇k.
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BibTeX
@misc{qu2026schur,
title = {{Schur positivity of the nabla operator on two-column modified Hall--Littlewood polynomials}},
author = {Menghao Qu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.20954},
eprint = {2605.20954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.20954},
}
RIS
TY - UNPB
TI - Schur positivity of the nabla operator on two-column modified Hall--Littlewood polynomials
AU - Menghao Qu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2605.20954
UR - https://arxiv.org/abs/2605.20954
AB - Bergeron, Garsia, Haiman, and Tesler (1999) introduced two Schur positivity conjectures concerning the action of the ∇ operator on modified Hall--Littlewood polynomials. Although the proofs of the celebrated compositional shuffle conjecture and its rational generalization, along with various symmetric function identities, have largely resolved these conjectures in the affirmative, we provide an elegant alternative proof driven by q-series techniques. Through this approach, we derive explicit Schur function expansions for two-column partitions and extend our results to arbitrary positive powers ∇^k.
ER -