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arXiv · 2610.39061

Bases for the Top-Degree Garsia-Haiman Module

Raymond Chou
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Abstract

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 $\widetilde{H}_μ(X;q,t)$. 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.

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Raymond Chou. 2026-10-28. Bases for the Top-Degree Garsia-Haiman Module. https://arxiv.org/abs/2610.39061

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