论文标题
麦克唐纳 - 赫维兹号
Macdonald-Hurwitz Number
论文作者
论文摘要
受到J. Novak的启发，关于BGW和HCIZ矩阵积分的渐近行为\ Cite {[n0]}以及hurwitz数字\ cite \ cite {[ip]}的代数和几何特性的启发。 \ cite {[z1]}，并通过相对GW-Invariants \ cite \ cite \ cite {[ip]}，\ cite {[lr]}的符号手术理论，使用来自整体macdonald函数的函数的函数到同源对中的功能和构建的函数的函数，使用构建函数的函数}麦克唐纳 - 赫维兹的数字。作为一种应用，我们已经构建了一系列新的属属剪切和加入差异操作员，可以将其视为Laplace-Beltrami运算符的概括，并以属属扩展的积分麦克唐纳函数作为其常见特征函数。我们还获得了一些相同程度的生成波函数，这些函数由麦克唐纳 - 赫维兹数字生成，可以用新的切割和加入差分差异操作员和初始值表示。另一个应用程序是，我们已经构建了一个新的交换缔合代数$（c（\ mathbb {f} [s_ {d}]）），\ circ_ {q，t}）$（请参考最后一个（6））。 通过沿特殊路径$η（a | b）$（参考公式（140），（141））的极限，我们专业化$（c（\ mathbb {f} [s_ {d}]），\ circ_ {q，t}）$ $（c（\ hat {\ mathbb {f}} [s_ {d}]），\ circ_ {a | b}）$，这将被证明是同构的 中等维的$ \ mathbb {\ mathbb {c}^*} $ - 通过杰克函数在$ \ mathbb {c}^2 $ W. li，Z.
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 $η(A|B)$ (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]}.}
