The nabla operator, Hall--Littlewood polynomials and LLT polynomials
Abstract
Bergeron, Garsia, Haiman, and Tesler conjectured that (−1)n−ℓ(μ)∇H̃μ[X;0,t] and ∇ωH̃μ[X;0,t−1] are Schur positive, and that ∇ωH̃μ[X;0,t−1]−∇ωH̃ν[X;0,t−1] is Schur positive when μ dominates ν. The first statement follows from the compositional shuffle conjecture of Haglund, Morse, and Zabrocki, proved by Carlsson and Mellit. We prove the other two statements. All three statements also hold for ∇r with r≥1. The main new step is an identity: when μ covers ν, the difference ωH̃μ[X;0,t−1]−ωH̃ν[X;0,t−1] is minus a power of t times one LLT polynomial. Using the Catalanimals and the Cauchy identity of Blasiak, Haiman, Morse, Pun, and Seelinger, we show that ∇r of this LLT polynomial is, up to sign, a positive sum of LLT polynomials. We also extend the main results to the Schiffmann algebra setting.
Related
- Schur positivity of the nabla operator on two-column modified Hall--Littlewood polynomials
- Reflection equations and Littlewood identities for spin Hall-Littlewood and q-Whittaker functions
- Changing parameters in Hall-Littlewood functions
- A finite cover for coefficient positivity of stretched Littlewood-Richardson polynomials in the seven-row, size-thirty box
- The norm of the centered Hardy--Littlewood maximal operator
BibTeX
@misc{gu2026nabla,
title = {{The nabla operator, Hall--Littlewood polynomials and LLT polynomials}},
author = {Feng Gu and Mingyang Kang and Dun Qiu and Minhao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16756},
eprint = {2610.16756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16756},
}
RIS
TY - UNPB
TI - The nabla operator, Hall--Littlewood polynomials and LLT polynomials
AU - Feng Gu
AU - Mingyang Kang
AU - Dun Qiu
AU - Minhao Zhang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16756
UR - https://arxiv.org/abs/2610.16756
AB - Bergeron, Garsia, Haiman, and Tesler conjectured that (−1)^n−ℓ(μ)∇H̃_μ[X;0,t] and ∇ωH̃_μ[X;0,t^−1] are Schur positive, and that ∇ωH̃_μ[X;0,t^−1]−∇ωH̃_ν[X;0,t^−1] is Schur positive when μ dominates ν. The first statement follows from the compositional shuffle conjecture of Haglund, Morse, and Zabrocki, proved by Carlsson and Mellit. We prove the other two statements. All three statements also hold for ∇^r with r≥1. The main new step is an identity: when μ covers ν, the difference ωH̃_μ[X;0,t^−1]−ωH̃_ν[X;0,t^−1] is minus a power of t times one LLT polynomial. Using the Catalanimals and the Cauchy identity of Blasiak, Haiman, Morse, Pun, and Seelinger, we show that ∇^r of this LLT polynomial is, up to sign, a positive sum of LLT polynomials. We also extend the main results to the Schiffmann algebra setting.
ER -