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Haglund--Haiman--Loehr formula via Carlsson--Mellit Algebra
Abstract
We prove a Haglund--Haiman--Loehr type combinatorial formula for the torus fixed point classes Iμ,w in the equivariant K-theory Kℂ*×ℂ*(PFHn,n−k) of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When w=∅, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson--Gorsky--Mellit action of the Carlsson--Mellit algebra 𝔸q,t.
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BibTeX
@misc{cho2026haglundhaimanloehr,
title = {{Haglund--Haiman--Loehr formula via Carlsson--Mellit Algebra}},
author = {Younggwang Cho and Jaeseong Oh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03840},
eprint = {2609.03840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03840},
}
RIS
TY - UNPB
TI - Haglund--Haiman--Loehr formula via Carlsson--Mellit Algebra
AU - Younggwang Cho
AU - Jaeseong Oh
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03840
UR - https://arxiv.org/abs/2609.03840
AB - We prove a Haglund--Haiman--Loehr type combinatorial formula for the torus fixed point classes I_μ,w in the equivariant K-theory K_ℂ^*×ℂ^*(PFH_n,n−k) of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When w=∅, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson--Gorsky--Mellit action of the Carlsson--Mellit algebra 𝔸_q,t.
ER -