稳定对称级数、微分算子与杰克形变
Stable Symmetric Series, Differential Operators, and Jack Deformations
AI总结：
该研究引入稳定对称级数，关联Ivanov-Kerov代数等数学对象，扩展至杰克多项式并验证其特化，为对称群相关算子研究提供新路径。
AI中文摘要：
我们引入稳定对称级数，该级数可同时对所有对称群编码标准化共轭类及其乘法算子。这为从Ivanov-Kerov代数到移位对称函数及$U(\boldsymbol{\textit{W}}_{1+\boldsymbol{\textit{\text{infty}}}})$中的微分算子提供了直接路径。利用Goulden-Jackson乘积，我们将该构造扩展至杰克多项式，恢复移位杰克特征值与皮耶里型关系，并获得次数为3和4的显式候选算子。其对扎布伦多项式的实部与四元数特化，通过高斯矩阵积分与穷举威克枚举得到验证。
英文摘要：
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.
