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arXiv · 2608.25651

Stable Symmetric Series, Differential Operators, and Jack Deformations

Jean-Yves Thibon
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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 $U(\mathcal W_{1+\infty})$. 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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Jean-Yves Thibon. 2026-08-26. Stable Symmetric Series, Differential Operators, and Jack Deformations. https://arxiv.org/abs/2608.25651

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