Congruences of shifted Jack Littlewood-Richardson coefficients
Abstract
The shifted Jack Littlewood-Richardson coefficients generalize the ordinary Jack coefficients and are Laurent polynomials in the Jack parameter α. We prove a previously conjectured congruence: coefficients indexed by triples differing by a single box move are congruent modulo the shared α-hook at the pivot. We also prove a shifted Macdonald analogue, with a power-of-t twist, and establish that the normalized shifted Macdonald coefficients are Laurent polynomials in q and t. The proofs combine coincidences of shifted coordinates with Laurent-preserving shift transforms, and the Macdonald input uses Knop's inversion formula and the integrality of the Bergeron-Garsia-Haiman-Tesler operators. Finally, we realize the Jack and Macdonald congruences as necessary edge conditions on Hilbert schemes of points, in equivariant cohomology and equivariant K-theory, respectively. In the Macdonald case a tautological determinant twist accounts for the power-of-t normalization
Related
- Positivity of stretched Littlewood-Richardson coefficients for partitions of length at most five
- A finite cover for coefficient positivity of stretched Littlewood-Richardson polynomials in the seven-row, size-thirty box
- Positivity for a t-analogue of refined Littlewood-Richardson rules
- A Littlewood-Richardson Rule for Forest Polynomials
- Characterization of k-highest weight and ĝ-dominant tableaux via $1−0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes
BibTeX
@misc{mickler2026congruences,
title = {{Congruences of shifted Jack Littlewood-Richardson coefficients}},
author = {Ryan Mickler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.17822},
eprint = {2606.17822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.17822},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Congruences of shifted Jack Littlewood-Richardson coefficients
AU - Ryan Mickler
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.17822
UR - https://arxiv.org/abs/2606.17822
AB - The shifted Jack Littlewood-Richardson coefficients generalize the ordinary Jack coefficients and are Laurent polynomials in the Jack parameter α. We prove a previously conjectured congruence: coefficients indexed by triples differing by a single box move are congruent modulo the shared α-hook at the pivot. We also prove a shifted Macdonald analogue, with a power-of-t twist, and establish that the normalized shifted Macdonald coefficients are Laurent polynomials in q and t. The proofs combine coincidences of shifted coordinates with Laurent-preserving shift transforms, and the Macdonald input uses Knop's inversion formula and the integrality of the Bergeron-Garsia-Haiman-Tesler operators. Finally, we realize the Jack and Macdonald congruences as necessary edge conditions on Hilbert schemes of points, in equivariant cohomology and equivariant K-theory, respectively. In the Macdonald case a tautological determinant twist accounts for the power-of-t normalization
ER -