移位麦克唐纳多项式与(q,t)变形的古尔德恩-杰克逊乘积
Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product
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中文总结 AI 辅助
本文研究移位麦克唐纳多项式与(q,t)变形古尔德恩-杰克逊乘积，构造实现移位麦克唐纳本征值的无穷级数，推导纳扎罗夫-斯克莱宁算子相关级数并与已有算子比较。
中文摘要 AI 辅助
在前期工作中，我们引入了稳定对称级数，该级数同时编码了所有对称群的标准化共轭类。这类有理级数对于杰克变形的古尔德恩-杰克逊乘积同样保持稳定性。我们研究其双参数麦克唐纳类似物：从(q,t)变形类乘积（与J基上的余积对角对偶）出发，构造出唯一的无穷级数，其乘法可实现任意移位麦克唐纳本征值。与经典情形和杰克情形不同，该变换不再是固定显式级数的乘法，我们将其识别为柯西乘法、积分nabla算子与简单对角算子的复合。随后确定实现纳扎罗夫-斯克莱宁算子A^(k)的级数，推导所有列分划的单一生成级数，并将我们的构造与本·达利和达达里奥的麦克唐纳特征标及西塔算子进行比较。
英文摘要
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.
发表机构
- Laboratoire d’Informatique Gaspard-Monge, Université Gustave Eiffel, CNRS, ESIEE Paris（加斯帕尔-蒙日信息实验室，巴黎高等理工学院，法国国家科学研究中心，巴黎ESIEE学院）
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