Shifted Macdonald Polynomials and the (q,t)-Deformed Goulden--Jackson Product
Abstract
In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden--Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the (q,t)-deformed class product (dual to the coproduct diagonal on the J-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov--Sklyanin operators A(k), derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D'Adderio.
Related
BibTeX
@misc{thibon2026shifted,
title = {{Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product}},
author = {Jean-Yves Thibon},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30791},
eprint = {2608.30791},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30791},
}
RIS
TY - UNPB
TI - Shifted Macdonald Polynomials and the (q,t)-Deformed Goulden--Jackson Product
AU - Jean-Yves Thibon
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30791
UR - https://arxiv.org/abs/2608.30791
AB - In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden--Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the (q,t)-deformed class product (dual to the coproduct diagonal on the J-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov--Sklyanin operators A^(k), derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D'Adderio.
ER -