PreprintCC BY 4.0arXiv
Bases for the Top-Degree Garsia-Haiman Module
Abstract
We introduce two new monomial bases for the top t-degree component of the Garsia-Haiman module Vμ for every partition μ. Both bases are indexed by column-strict standard fillings of shape μ; the first has exponents governed by entrywise refinements of Haglund's combinatorial statistics for the modified Macdonald polynomials H̃μ(X;q,t). The second basis has exponents governed by an insertion algorithm of Bhattacharya-Ratheesh-Viswanath, and gives a basis realization of the ``fermionic formula" of the m1n-coefficient of the q-Whittaker polynomial. Our proof uses orbit harmonics, and is inspired by the original approach of Garsia and Haiman.
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BibTeX
@misc{chou2026bases,
title = {{Bases for the Top-Degree Garsia-Haiman Module}},
author = {Raymond Chou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39061},
eprint = {2610.39061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39061},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bases for the Top-Degree Garsia-Haiman Module
AU - Raymond Chou
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.39061
UR - https://arxiv.org/abs/2610.39061
AB - We introduce two new monomial bases for the top t-degree component of the Garsia-Haiman module V_μ for every partition μ. Both bases are indexed by column-strict standard fillings of shape μ; the first has exponents governed by entrywise refinements of Haglund's combinatorial statistics for the modified Macdonald polynomials H̃_μ(X;q,t). The second basis has exponents governed by an insertion algorithm of Bhattacharya-Ratheesh-Viswanath, and gives a basis realization of the ``fermionic formula" of the m_1^n-coefficient of the q-Whittaker polynomial. Our proof uses orbit harmonics, and is inspired by the original approach of Garsia and Haiman.
ER -