Macdonald-Hurwitz Number

TL;DR

This paper introduces Macdonald-Hurwitz numbers, constructs new differential operators and generating functions related to symmetric functions, and establishes a connection with cohomological rings of Hilbert schemes, advancing algebraic and geometric understanding.

Contribution

It constructs Macdonald-Hurwitz numbers, develops genus-expanded cut-and-join operators, and links these to cohomological rings of Hilbert schemes, providing new algebraic structures and insights.

Findings

Construction of Macdonald-Hurwitz numbers

Development of genus-expanded differential operators

Establishment of algebraic isomorphism with cohomological rings

Abstract

Inspired by J. Novak's works on the asymptotic behavior of the BGW and the HCIZ matrix integrals \cite{[N0]} and by the algebraic and geometric properties of the Hurwitz numbers \cite{[IP]}, \cite{[LZZ]}, \cite{[LR]}, \cite{[OP]}, \cite{[Z1]}, and by the symplectic surgery theory of the relative GW-invariants \cite{[IP]}, \cite{[LR]}, using the elements of the transform matrix from the integral Macdonald function with two parameters to the homogeneous symmetric power sum functions \cite{[M]}, we have constructed the Macdonald-Hurwitz numbers. As an application, we have constructed a series of new genus-expanded cut-and-join differential operators, which can be thought of as the generalization of the Laplace-Beltrami operators and have the genus-expanded integral Macdonald functions as their common eigenfunctions. We have also obtained some generating wave functions of the same degree, which are generated by the Macdonald-Hurwitz numbers and can be expressed in terms of the new cut-and-join differential operators and the initial values. Another application is that we have constructed a new commutative associative algebra $(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ (referring to the last section (6)). By taking the limit along a special path $\eta(A|B)$ (referring to the formulas (140), (141)), we specialize $(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ to be a commutative associative algebra $(C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})$, which will be proven to be isomorphic to the middle-dimensional $\mathbb{\mathbb{C}^*}$-equivalent cohomological rings via the Jack functions over the Hilbert scheme points of $\mathbb{C}^2$ constructed by W. Li, Z. Qin, and W. Wang in \cite{[LQW2]}.}