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Stable Symmetric Series, Differential Operators, and Jack Deformations
Abstract
We introduce stable symmetric series which encode normalized conjugacy classes and their multiplication operators simultaneously for all symmetric groups. This gives a direct route from the Ivanov--Kerov algebra to shifted symmetric functions and to differential operators in U(𝒲1+∞). Using the Goulden--Jackson product, we extend the construction to Jack polynomials, recover shifted Jack eigenvalues and Pieri-type relations, and obtain explicit candidate operators in degrees three and four. Their real and quaternionic specializations to zonal polynomials are verified by Gaussian matrix integrals and exhaustive Wick enumeration.
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BibTeX
@misc{thibon2026stable,
title = {{Stable Symmetric Series, Differential Operators, and Jack Deformations}},
author = {Jean-Yves Thibon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25651},
eprint = {2608.25651},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25651},
}
RIS
TY - UNPB
TI - Stable Symmetric Series, Differential Operators, and Jack Deformations
AU - Jean-Yves Thibon
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2608.25651
UR - https://arxiv.org/abs/2608.25651
AB - We introduce stable symmetric series which encode normalized conjugacy classes and their multiplication operators simultaneously for all symmetric groups. This gives a direct route from the Ivanov--Kerov algebra to shifted symmetric functions and to differential operators in U(𝒲_1+∞). Using the Goulden--Jackson product, we extend the construction to Jack polynomials, recover shifted Jack eigenvalues and Pieri-type relations, and obtain explicit candidate operators in degrees three and four. Their real and quaternionic specializations to zonal polynomials are verified by Gaussian matrix integrals and exhaustive Wick enumeration.
ER -