Macdonald-Hurwitz Number
Quan Zheng 111E-mail<EMAIL_ADDRESS>
Mathematics College, Sichuan University, 610064,

Chengdu, Sichuan, PRC

November 11, 2022

Abstract

Inspired by J. Novak’s works on the asymptotic behavior of the BGW and the
HCIZ matrix integrals [32] and by the algebraic and geometric properties of
the Hurwitz numbers [16], [22], [24], [34], [39], and by the symplectic
surgery theory of the relative GW-invariants [16], [24], using the elements of
the transform matrix from the integral Macdonald function with two parameters
to the homogeneous symmetric power sum functions [28], 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 [26].

Keyword: Macdonald-Hurwitz number; integral Macdonald function; cut-and-join
operator; generating wave function; $\mathbb{C}^{*}$-equivalent cohomological
ring
2020 Mathematics Subject Classification: 14N10, 53D45, 53Z05

###### Contents

  1. 1 Introduction
  2. 2 Notations and Preliminary
    1. 2.1 Partitions
    2. 2.2 Symmetric function
    3. 2.3 Macdonald functions with two parameters $(q,t)$
    4. 2.4 The Integral Macdonald functions with two parameters $(q,t)$
    5. 2.5 The initial values
  3. 3 Macdonald-Hurwitz number
  4. 4 The genus-expanded cut-and-join operators/ integral Macdonald functions with two parameters $(q,t)$

##  1\. Introduction

Our primitive motivation for the paper is to understand and generalize J.
Novak’s works about the asymptotic behavior of the BGW and the HCIZ matrix
integrals [32]. To do this, our first step is to set up a link between the
integral Macdonald functions with the theory of the Riemann surfaces by
applying the Macdonald-Hurwitz numbers defined in the paper, referring to the
definition (3.1). In our views, the asymptotic expansions of the BGW and the
HCIZ matrix integrals are actually related to the theory of the Riemann
surface, because these integrals can be expressed by the Hurwitz number [15]
or the Schur functions, both of which have the orthogonality [28], [39]. More
generally, if an integral can be expressed by the Jack functions with a Jack
parameter $\alpha$ (including the zonal functions $\alpha=2$, the Schur
functions $\alpha=1,$ and the symplectic zonal functions
$\alpha=\frac{1}{2}$), or the (integral) Macdonald functions with two
parameters $q,t,$ [28], then the asymptotic expansions should be actually
related to the theory of the Riemann surfaces due to the Macdonald-Hurwitz
numbers, or to the orthogonality of those functions, seeing [28] or the
orthogonal/cutting lemma (2.3) in the paper. We plan to further investigate
this in our next step.

If we say that the differential operators, eigenfunctions, and eigenvalues are
the three musketeers playing important roles in mathematics and physics, then
the classical cut-and-join operators [10], [11], [12] and the Schur functions
[28] are the pair of rangers who are playing a stage drama of love and hate.
They are flashing from time to time in the classical [15] or all kinds of
refined (such as [7], [9]) Hurwitz Enumeration Problem and the integrable
systems, the Hodge integral, the harmonic analysis in the Lie group, etc [10],
[11], [12], [22], [34]. Meanwhile, the Hurwitz theory serves as a great
matchmaker to connect the theory of the Riemann surface with the theory of the
BGW and the HCIZ integrals [13], [17]. The people pursue the topology/genus
expansion property of a differential operator or a function, which is
demonstrated clearly and vividly in the study of the limit of the BGW
integrals and the HCIZ integrals [32], [33]. We will continue to adhere to
this principle in this paper.

Inspired by the algebraic expressions of the Hurwitz numbers, which are mainly
expressed by the elements of the transform matrix from the Schur functions
$s_{\lambda}$ to the homogeneous symmetric power sum functions $p_{\lambda}$,
and by the symplectic surgery theory of the relative GW-invariants developed
by A.M. Li and Y.B. Ruan [24], E. Ionel and T. Parker [16], using the elements
of transform matrix from the integral Macdonald functions $J_{\lambda}(x;q,t)$
with two parameters $(q,t)$ [28] to the homogeneous symmetric power sum
functions $p_{\lambda}$, we have constructed the Macdonald-Hurwitz numbers. As
an application, we obtain the new genus-expanded cut-and-join operators (90)
similar to [2], [30], [39],[40], which have the genus-expanded integral
Macdonald functions (LABEL:11) as the common eigenfunctions. The genus-
expanded cut-and-join operators can be thought of as the generalization of the
Laplace-Beltrami operators.

As a special case, let us see what has happened to the zonal polynomials [20],
[21], [38]. From the view of the geometry, it is well known that the zonal
polynomials are eigenfunctions of the Laplace-Beltrami operator [14], [21]

(1) $\Delta=\frac{1}{det(g_{ij})}\sum_{i,j=1}^{N}\frac{\partial}{\partial
x_{i}}\sqrt{det(g_{ij})}g^{ij}\frac{\partial}{\partial x_{j}},$

where $x_{1},x_{2},\cdots,x_{N}$ are coordinates of a point in a space $M$
with Riemannian metric differential form

(2) $\displaystyle(ds)^{2}=\sum_{i,j=1}^{N}g_{ij}dx_{i}dx_{j}$ (3)
$\displaystyle(g^{ij})=(g_{ij})^{-1}$

Especially, let $GL(N,{\mathbb{R}}),O(N,{\mathbb{R}})$ be the real general
linear group and the real orthogonal linear group, then we consider the
homogenous space $M\cong GL(N,{\mathbb{R}})/O(N,{\mathbb{R}})$, which is
isomorphic to the space of $N\times N$ real positive definite symmetric
matrices $X$ (by sending a nonsingular matrix $V$ to $VV^{\prime}$) with
congruence invariant Riemannian metric [21], [29]

(4) $\displaystyle(ds)^{2}=tr(X^{-1}dXX^{-1}dX).$

Then the Laplace-Beltrami operator concerned with the latent root
$x_{1},\cdots,x_{N}$ of $X$ is the following [21]

(5) $\hat{D}:=\sum_{i=1}^{N}x_{i}^{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}x_{i}^{2}(x_{i}-x_{j})^{-1}\frac{\partial}{\partial
x_{j}}-\sum_{i=1}^{N}\frac{1}{2}(N-3)x_{i}\frac{\partial}{\partial x_{i}}$

where the last term will not change a homogeneous polynomial as eigenfunction
but the eigenvalue, hence we leave this term out as [6], [7], [8]

(6) $D:=\sum_{i}^{N}x_{i}^{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}x_{i}^{2}(x_{i}-x_{j})^{-1}\frac{\partial}{\partial x_{j}}$

In view of physics, the zonal polynomials are the wave functions and the
eigenvalues are the energy eigenvalues [6]. To deal with the
$Schr\ddot{o}dinger$ equations for the one-dimensional N-body problem, the
physicist F. Calogero [6] proposed a similar Laplace-Beltrami operator as the
formula (5) in 1969.

Let us introduce the time-variables or Miwa variables $p_{1},p_{2},\cdots,$
(the power sum of the laten roots $x_{1},\cdots,x_{N}$ of $X$), i.e., for any
positive integer $i\geq 1$,

(7) $p_{i}=\sum_{j=1}^{N}x_{j}^{i},$

then the Laplace-Beltrami operator (6) acting on the functions in the time-
variables such as $f(p_{1},p_{2},\cdots)$ is equivalent to the classical cut-
and-join operator with partition type $(1^{d-2}2)$ [10],[22],[30] as

(8) $\displaystyle D=\sum_{k,l\geq
1}\left[klp_{k+l}\frac{\partial^{2}}{\partial p_{k}\partial
p_{l}}+(k+l)p_{k}p_{l}\frac{\partial}{\partial p_{k+l}}\right]$

up a multiple $\frac{1}{2}$, which will again not change the eigenfunctions
but eigenvalues.

Meanwhile, the zonal polynomial is closely concerned with the irreducible
character of the general real linear group $GL(N,\mathbb{R})$, [19], [28].

In this paper, we have extended the above theory to the genus-expanded cut-
and-join operators and the genus-expanded integral Macdonald functions by
defining the Macdonald-Hurwitz numbers, such that the latter is the common
eigenfunctions of the former. Let $d\geq 0$ be non-negative integer, and
$\lambda,\Delta$ be the partition of $d,$ which is denoted by
$\lambda,\Delta\vdash d.$ Let $J_{\lambda}(x;q,t)$ be the integral Macdonald
functions with two parameters $q,t,$ [28], which can be expressed by the power
sum functions $p_{\Delta}$, say, [28]

(9) $J_{\lambda}(x;q,t)=\sum_{\Delta\vdash
d}a_{\lambda}(\Delta)(q,t)p_{\Delta},$

where

(10) $p_{\Delta}=\prod_{i}p_{\delta_{i}}$

if $\Delta=(\delta_{1},\delta_{2},\cdots),$ and
$a_{\lambda}(\Delta)(q,t)\in{\mathbb{F}}:=\mathbb{Q}(q,t),$ the field of the
rational function with the rational coefficients in $q,t,$ especially, we
denote $dim\lambda:=a_{\lambda}(1^{|\lambda|})(q,t).$

Let $\Sigma^{h}$ be a closed connected Riemann surface of genus $h\geq 0,$ and
$\Sigma^{g}$ a closed connected Riemann surface of genus $g\geq 0$. For a
general given points set $\\{q_{1},\cdots,q_{k}\\}\in\Sigma^{h}$, which is
called the set of branch points, we call a holomorphic map
$f:\Sigma^{g}\to\Sigma^{h}$ a ramified covering of $\Sigma^{h}$ of degree
$d\geq 0$ by $\Sigma^{g}$ with a ramification type
$(\Delta_{1},\cdots,\Delta_{k})$, if the preimages of
$f^{-1}(q_{i})=\\{p_{i}^{1},\cdots,p_{i}^{l_{i}}\\}$ with orders
$\Delta_{i}=(\delta^{i}_{1},\cdots,\delta^{i}_{l_{i}})\vdash d$ for
$i=1,\cdots,k,$ respectively, where $l_{i}=l(\Delta_{i})$ is the length of the
partition $\Delta_{i}.$

######  Definition 1.1.

(Definition 3.1) We define the connected genus $g\geq 0$ Macdonald-Hurwitz
number with degree $d>0$ over the closed connected Riemann surface
$\Sigma^{h}$ with the ramification type $(\Delta_{1},\cdots,\Delta_{k})$ as

(11) $\displaystyle MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})=$
$\displaystyle\sum_{\lambda\vdash
d}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{1-h}\frac{a_{\lambda}(\Delta_{1})(q,t)}{dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{k})(q,t)}{dim\lambda},$

under the condition

(12) $(2-2h)d-(2-2g)=\sum_{i=1}^{k}(d-l(\Delta_{i})),$

otherwise, we define

(13) $\displaystyle MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})=0,$

and if we take $d=0,$ it is convenient to define

(14) $\displaystyle MH_{h}^{g,0}(\Delta_{1},\cdots,\Delta_{k})=0.$

where $j_{\lambda}(q,t)$ is defined by the formula (48).

Taking the rule of the exponential-logarithmic correspondence, we can define
the disconnected Macdonald-Hurwitz numbers, which are obtained by taking the
product of all connected the Macdonald-Hurwitz numbers module their
automorphisms, and then summing up all possibilities, as described in section
(3).

We can iteratively calculate the Macdonald-Hurwitz numbers by decreasing the
genus of the target surface and the numbers of the branch points,

######  Theorem 1.2.

(Theorem3.3)

  * (1)

Decrease the genus of the target surface directly, i.e., if $h\geq 1,$ we have

(15)
$\displaystyle{}MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})=\sum_{\Delta\vdash
d}MH_{h-1}^{g,d}(\Delta_{1},\cdots,\Delta_{k},\Delta,\Delta)z_{\Delta}(q,t),$

where $z_{\Delta}(q,t)$ is defined by the formula (43);

  * (2)

(Associativity) Decrease the genus of the target surface and the numbers of
the branched points by cutting it into two parts, i.e, for
$h_{1}+h_{2}=h,h_{1}\geq 0,h_{2}\geq 0,$ and any positive integer $k>l\geq 1$,
we have

$\displaystyle MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})$ $\displaystyle=$
$\displaystyle\sum_{\Delta}MH_{h_{1}}^{g_{1},d}(\Delta_{1},\cdots,\Delta_{l},\Delta)z_{\Delta}(q,t)$
$\displaystyle\times
MH_{h_{2}}^{g_{2},d}(\Delta,\Delta_{l+1},\cdots,\Delta_{k}),$

where the genus $g,g_{1},g_{2}$ are determined by the formula
$(\ref{hurwitz})$.

######  Remark 1.3.

We have noticed the following interesting facts (see Corollary (3.4) and
Remark (3.5) and the formula (43)):

(17) $MH_{0}^{0,1}((1),(1))=\frac{1}{z_{(1)}(q,t)}=\frac{1-t}{1-q},$ (18)
$MH_{0}^{0,1}((1))=\frac{1-t}{1-q},$

thus we can determine $\frac{1-t}{1-q}$-curves passing through two points or
one point.

Let $\hbar$ be indeterminate, we have

######  Definition 1.4.

(Definition 4.1) Following the idea of [39], for any partition $\Delta\vdash
d$, we define a genus-expanded cut-and-join operator $D(\Delta,\hbar)$
(referring to the formula (90))

(19) $D(\Delta,\hbar)=\sum_{\Gamma,\Gamma^{\prime}\vdash
d}z_{\Gamma^{\prime}}(q,t)\hbar^{d+l(\Gamma^{\prime})-l(\Delta)-l(\Gamma)}MH_{0}^{g,d}(\Gamma^{\prime},\Delta,\Gamma)p_{\Gamma}\frac{\partial}{\partial
p_{\Gamma^{\prime}}},$

where genus $g$ satisfies:

(20) $2d-(2-2g)=(d-l(\Gamma^{\prime}))+(d-l(\Delta))+(d-l(\Gamma)),$

and $\frac{\partial}{\partial p_{\Gamma^{\prime}}}$ is defined by the formula
(38).

Let $J_{\lambda}(x;q,t,\hbar)$ be the genus-expanded integral Macdonald
functions (referring to the formula (LABEL:11)), we have

######  Theorem 1.5.

(Theorem LABEL:eigen) For any partition $\Delta,\lambda\vdash d,$ the
following equality holds:

(21)
$D(\Delta,\hbar)J_{\lambda}(x;q,t,\hbar)=\hbar^{d-l(\Delta)}a_{\lambda}(\Delta)(q,t)J_{\lambda}(x;q,t,\hbar),$

i.e., the genus-expanded integral Macdonald functions
$J_{\lambda}(x;q,t,\hbar)$ are the common eigenfunctions of the genus-expanded
cut-and-join operators $D(\Delta,\hbar)$ and
$\hbar^{d-l(\Delta)}a_{\lambda}(\Delta)(q,t)$ are the corresponding
eigenvalues.

For a given $d$, the genus-expanded cut-and-join operators, which act on the
functions in the time-variables $p=(p_{1},p_{2},\cdots)$, are closed under the
multiplication, i.e.,

######  Theorem 1.6.

(Theorem LABEL:operator)For any partitions $\Delta_{1},\Delta_{2}\vdash d,$ we
have

(22) ${D(\Delta_{1},\hbar)D(\Delta_{2},\hbar)=\sum_{\Delta_{3}\vdash
d}\hbar^{d-l(\Delta_{1})-l(\Delta_{2})+l(\Delta_{3})}C_{\Delta_{1}\Delta_{2}}^{\Delta_{3}}D(\Delta_{3},\hbar)},$

where

(23)
$C_{\Delta_{1}\Delta_{2}}^{\Delta_{3}}=z_{\Delta_{3}}(q,t)MH_{d}^{g}(\Delta_{1},\Delta_{2},\Delta_{3})$

is a new structure constant of the center subalgebra $C(\mathbb{F}[S_{d}])$ of
the group algebra $\mathbb{F}[S_{d}]$ defined by the formula (LABEL:constant).

######  Remark 1.7.

The terminology "new" in the paper is relative to the classical Hurwitz theory
[30], [34], [39], etc.

Let
$\Phi_{h}\\{\hbar|(u_{1},\Delta_{1}),\cdots,(u_{n},\Delta_{n})|p^{(1)},\cdots,p^{(k)},p\\}$
be the generating wave function (referring to the formula (LABEL:99)), then
for any $i\in\\{1,\cdots,n\\},$ we have

######  Theorem 1.8.

(Theorem LABEL:de)

(24)
$\displaystyle\frac{\partial\Phi_{h}\\{\hbar|(u_{1},\Delta_{1}),\cdots,(u_{n},\Delta_{n})|p^{(1)},\cdots,p^{(k)},p\\}}{\partial
u_{i}}$ $\displaystyle=$ $\displaystyle
D(\Delta_{i},\hbar)\Phi_{h}\\{\hbar|(u_{1},\Delta_{1}),\cdots,(u_{n},\Delta_{n})|p^{(1)},\cdots,p^{(k)},p\\}.$

Furthermore, let $S_{d}$ be the symmetric group of the degree $d,$ then we
have applied the Macdonald-Hurwitz numbers to determine a
$\circ_{q,t}$-product on the central subalgebra $C(\mathbb{F}[S_{d}])$ of the
group algebra $\mathbb{F}[S_{d}]$, which can be thought of as a
$(q,t)$-deformation product of $C(\mathbb{C}[S_{d}])$ if we extend the
coefficient field $\mathbb{C}$ to the rational functions fields
$\mathbb{F}:=\mathbb{Q}(q,t)$, as described in the last section (6),i.e. we
have

######  Theorem 1.9.

(Theorem LABEL:ca) For any partition $\Delta\vdash d,$ let $C_{\Delta}\in
C(\mathbb{C}[S_{d}])$ be the classical central element of
$C(\mathbb{C}[S_{d}])$, then $\\{C_{\Delta}|\Delta\in{\mathbb{P}}_{d}\\}$ also
forms a ${\mathbb{F}}$-basis of $C(\mathbb{F}[S_{d}])$. By linearly extending,
$C(\mathbb{F}[S_{d}])$ is a new commutative associative algebra with the unit
$C_{(1^{d})}$ under the following product,

(25)
$C_{\Delta_{1}}\circ_{q,t}C_{\Delta_{2}}=C_{\Delta_{1}\Delta_{2}}^{\Delta_{3}}C_{\Delta_{3}},$

where

(26)
$C_{\Delta_{1}\Delta_{2}}^{\Delta_{3}}=z_{\Delta_{3}}(q,t)MH_{d}^{g}(\Delta_{1},\Delta_{2},\Delta_{3})$

is the new structure constant.

######  Remark 1.10.

In a series of the papers [3], [4], [35], [36], [37], the authors of those
papers have constructed the local Gromov-Witten invariants of the curves,
Donaldson-Thomas invariants, the quantum cohomology of the rank 2 bundles over
the Riemann surfaces or the Hilbert scheme points of $\mathbb{C}^{2}$, the
orbifold quantum cohomology of the symmetric product of $\mathbb{C}^{2}$, the
equivariant version of those invariants, which are proven to be equivalent and
closely related with the Jack functions and the Macdonald functions, it is an
interesting problem to research the relationships between those invariants
with our Macdonald-Hurwitz numbers, and we hope to come back to pursue the
problem in the future. The author would like to thank an anonymous expert for
his communications about the above information. However, by taking the limit
along a special path $\eta(A|B)$ (the formula (140), (141))), (for the other
notations and the terminologies referring to the last section (6)), we
specialize $(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ to a commutative associative
algebra $(C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})$ with the unit, 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 [26].

##  2\. Notations and Preliminary

Throughout this paper, we shall adopt the same notations as I. G. Macdonald’s
book [28]. Those experts who are familiar with the book can skip this section.

###  2.1. Partitions

A partition [28], p1, is a (finite or infinite) sequence

(27) $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{i},\cdots),$

of non-negative integers in decreasing order:

(28) $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{i}\geq\cdots\geq 0,$

and containing only finitely many nonzero terms. The number of the nonzero
terms is called the length of $\lambda$, denoted by $l(\lambda)$. The weight
$|\lambda|$ of $\lambda$ is defined by

(29) $|\lambda|:=\sum_{i}\lambda_{i}.$

It is convenient to denote the partition by

(30) $\lambda=(1^{m_{1}}2^{m_{2}}\cdots r^{m_{r}}\cdots),$

where $m_{i}=m_{i}(\lambda):=\\#\\{\lambda_{j}\in\lambda|\lambda_{j}=i\\}$ is
called the multiplicity of $i$ in $\lambda.$ Moreover we define [28], p24

(31) $z_{\lambda}=\prod_{i\geq 1}i^{m_{i}}.m_{i}!$

If $|\lambda|=d,$ we say that $\lambda$ is a partition of $d,$ and denote it
by $\lambda\vdash d.$ The set of all partitions of $d$ is denoted by
${\mathbb{P}}_{d},$ and the set of all partitions (including $d=0$) by
${\mathbb{P}},$ which has a natural lexicographic ordering, [28], p6, i.e.,
for any two partition $\lambda,\mu\in{\mathbb{P}}$ we call $\lambda\geq\mu$
iff the first non-vanishing difference $\lambda_{i}-\mu_{i}$ is non-negative.

The Ferrers diagram of a partition $\lambda$ may be formally defined as the
set of points $s=(i,j)\in{\mathbb{Z}}^{2}$ such that $1\leq i\leq
l(\lambda),1\leq j\leq\lambda_{i}$. The conjugate of a partition $\lambda$ is
the partition $\lambda^{\prime}$ whose diagram is the transpose of the diagram
$\lambda$, i.e. the diagram obtained by reflection in the main diagonal. Hence
$\lambda_{i}^{\prime}$, is the number of nodes in the $i$-th column of
$\lambda$, or equivalently

(32) $\lambda^{\prime}_{i}=\\#\\{j:\lambda_{j}\geq i\\}.$

We will also denote the Ferrers diagram by $\lambda.$ For each
$s=(i,j)\in\lambda,$ we define the arm-length $a(i,j)$, leg-length $l(i,j)$
[28], p337 and hook length $hook(i,j)$ [28], p10 by

(33) $\displaystyle a(s)=a(i,j)=\lambda_{i}-j,$ (34) $\displaystyle
l(s)=l(i,j)=\lambda^{\prime}_{j}-i,$ (35) $\displaystyle
h(s)=h(i,j)=a(i,j)+l(i,j)+1.$

Let $p=(p_{1},p_{2},p_{3},\cdots,)$ be indeterminate variables or the
homogeneous symmetric power sums (see, section(1)), which are called the time-
variables or Miwa variables, and we assume that
$\Delta=(\delta_{1},\cdots,\delta_{n})$ is a partition with length $n$. We
denote

(36) $\displaystyle\Delta!:=\prod_{i\geq 1}m_{i}(\Delta)!,$ (37)
$\displaystyle p_{\Delta}:=p_{\delta_{1}}\cdots p_{\delta_{n}},$ (38)
$\displaystyle\frac{\partial}{\partial
p_{\Delta}}:=\frac{1}{\Delta!}\frac{\partial}{\partial
p_{\delta_{1}}}\cdots\frac{\partial}{\partial p_{\delta_{n}}}.$

###  2.2. Symmetric function

The symmetric group $S_{N}$ acts the ring ${\mathbb{Q}}[x_{1},\cdots,x_{N}]$
of polynomials in $N$ independent variables $x=(x_{1},\cdots,x_{N})$ with
rational coefficients by permuting the variables, and a polynomial is
symmetric if it is invariant under this action. The symmetric polynomials form
a graded subring:

(39) $\displaystyle\Lambda_{N}$ $\displaystyle=$
$\displaystyle{\mathbb{Q}}[x_{1},\cdots,x_{N}]^{S_{N}}$ $\displaystyle=$
$\displaystyle\oplus_{d\geq 0}\Lambda_{N}^{d},$

where $\Lambda_{N}^{d}$ consists of the homogeneous symmetric polynomial of
degree $d\geq 0.$

It is well-known that [28], p18, p63, in $\Lambda^{d}_{N},$ the basis
consisted by the monomial symmetric functions
$\\{m_{\lambda}(x_{1},\cdots,x_{N})|\lambda\in{\mathbb{P}}_{d}\\}$ form a
$\mathbb{Z}$-basis, and the element symmetric polynomials
$\\{e_{\lambda}(x_{1},\cdots,x_{N})|\lambda\in{\mathbb{P}}_{d}\\}$ is dual to
the basis consisted by the complete symmetric polynomials
$\\{h_{\lambda}(x_{1},\cdots,x_{N})|\lambda\in{\mathbb{P}}_{d}\\},$ the basis
consisted by the power sum symmetric polynomials
$\\{p_{\lambda}(x_{1},\cdots,x_{N})|\\\ \lambda\in{\mathbb{P}}_{d}\\}$ form an
orthogonal basis of $\Lambda_{N}^{d}$, [28], p64, and the basis consisted by
the Schur symmetric polynomials
$\\{s_{\lambda}(x_{1},\cdots,x_{N})|\lambda\in{\mathbb{P}}_{d}\\}$ form an
orthonormal basis of $\Lambda_{N}^{d}$. In the rest of the paper, we will
neglect the variables $(x_{1},\cdots,x_{N})$, and write the above symmetric
polynomial as $m_{\lambda},e_{\lambda},h_{\lambda},p_{\lambda},s_{\lambda}.$

Taking the inverse limit [28], p18, we have $\mathbb{Z}$-modules
$\Lambda^{d}:$

(40) $\Lambda^{d}=\lim_{\leftarrow_{N}}\Lambda_{N}^{d}.$

Let

(41) $\Lambda=\oplus_{d\geq 0}\Lambda^{d},$

so that $\Lambda$ is a free graded $\mathbb{Z}$-module generated by the
monomial symmetric polynomial $m_{\lambda}$ for all partitions $\lambda$.
Moreover, the graded ring $\Lambda$ thus defined is called the ring of
symmetric functions in countably many independent variables $x_{1},x_{2},$ … .
For any commutative ring or field $\mathcal{A}$, we define

(42) $\Lambda_{\mathcal{A}}\cong\Lambda\otimes_{{\mathbb{Z}}}\mathcal{A}.$

Thus by the definition (37), $\\{p_{\Delta}|\Delta\in{\mathbb{P}}\\}$ form a
${\mathbb{Q}}-$ basis of $\Lambda_{\mathbb{Q}}.$

###  2.3. Macdonald functions with two parameters $(q,t)$

Let $q,t$ be independent variables, and let $\mathbb{F}={\mathbb{Q}}(q,t)$ be
the field of rational functions in $q$ and $t$, and let
$\Lambda_{\mathbb{F}}\cong\Lambda\otimes_{{\mathbb{Z}}}{\mathbb{F}}$ denote
the $\mathbb{F}$-algebra of symmetric functions with coefficients in
$\mathbb{F}$. We define $(q,t)$-deformation of the formula (31) by [28], p309,
(2.1)

(43)
$z_{\lambda}(q,t)=z_{\lambda}\prod_{i=1}^{l(\lambda)}\frac{{1-q^{\lambda_{i}}}}{{1-t^{\lambda_{i}}}}.$

Then the Macdonald symmetric functions $P_{\lambda}(x;q,t)$ depending
rationally on two parameters $(q,t)$ are characterized by the following two
properties [27], [28], p322

  * (a)

There exist $u_{\lambda\mu}\in{\mathbb{F}}$ such that

(44)
$P_{\lambda}(x;q,t)=m_{\lambda}+\sum_{\mu\textless\lambda}u_{\lambda\mu}m_{\mu},$

i.e., the transition matrix that expresses the Macdonald functions in terms of
the monomial symmetric functions is a strictly upper unit triangular matrix.

  * (b)

The Macdonald functions $P_{\lambda}(x;q,t)$ are pairwise orthogonal relative
to the scalar product defined by

(45) $<p_{\lambda},p_{\mu}>_{q,t}=\delta_{\lambda\mu}{{z}}_{\lambda}(q,t).$

###  2.4. The Integral Macdonald functions with two parameters $(q,t)$

For each partition $\lambda$, we denote [28], p352, (8.1), (8.1′),

(46) $\displaystyle
c_{\lambda}(q,t)=\prod_{s\in\lambda}(1-q^{a(s)}t^{l(s)+1}),$ (47)
$\displaystyle
c_{\lambda}^{\prime}(q,t)=\prod_{s\in\lambda}(1-q^{a(s)+1}t^{l(s)}),$ (48)
$\displaystyle j_{\lambda}(q,t)=c_{\lambda}(q,t)c_{\lambda}^{\prime}(q,t).$

taking $q=t^{\alpha},$ and letting $t\to 1$, we define [28], p381, (10.21),
(10.23),

(49) $\displaystyle c_{\lambda}^{(\alpha)}=\lim_{t\to
1}\frac{c_{\lambda}(t^{\alpha},t)}{(1-t)^{|\lambda|}}=\prod_{s\in\lambda}(\alpha
a(s)+l(s)+1),$ (50) $\displaystyle c_{\lambda}^{\prime(\alpha)}=\lim_{t\to
1}\frac{c_{\lambda}^{\prime}(t^{\alpha},t)}{(1-t)^{|\lambda|}}=\prod_{s\in\lambda}(\alpha(a(s)+1)+l(s)),$
(51) $\displaystyle j^{(\alpha)}_{\lambda}=\lim_{t\to
1}\frac{j_{\lambda}(q,t)}{(1-t)^{2|\lambda|}}=\prod_{s\in\lambda}(\alpha
a(s)+l(s)+1)(\alpha(a(s)+1)+l(s)).$

######  Remark 2.1.

In general, $j_{\lambda}(q,t)$ is not a perfect square except that $q=t,$ thus
we could not use the square root of $j_{\lambda}(q,t)$ to define the
Macdonald-Hurwitz number because we hope that the Macdonald-Hurwitz numbers
are the rational functions in the parameters $q,t,$ (see the definition 3.1).

######  Definition 2.2.

([28], p352, (8.3)) The integral Macdonald functions with two parameter
$(q,t)$ is defined by

(52) $\displaystyle J_{\lambda}(x;q,t)=c_{\lambda}(q,t)P_{\lambda}(x;q,t).$

Then we have the following orthogonal lemma, which is a key point to our
paper,

######  Lemma 2.3.

the Orthogonal Lemma/Cutting Lemma [28], p353, (8.7); p357. The integral
Macdonald functions $J_{\lambda}(x;q,t)$ have scalar products relative to the
scalar product of the formula (45)

(53)
$<J_{\lambda}(x;q,t),J_{\mu}(x;q,t)>_{q,t}=\delta_{\lambda\mu}j_{\lambda}(q,t).$

######  Remark 2.4.

The terminology "cutting" comes from the symplectic surgery theory, for the
detail, referring to [22], [24].

We list two important special cases of the integral Macdonald functions which
are mostly related to our paper:

  * (1)

When $q=t$,

(54) $J_{\lambda}(x;t,t)=s_{\lambda}\prod_{(i,j)\in\lambda}(1-t^{hook(i,j)}),$

where $s_{\lambda}$ is the Schur function.

  * (2)

Taking $q=t^{\alpha}$, and letting $t\to 1,$ we arrive at the Jack symmetric
functions, which were first defined by the statistician Henry Jack in 1969,
[18], [28], p353, p381, (10.22),

(55) $J^{(\alpha)}_{\lambda}(x)=\lim_{t\to
1}\frac{J_{\lambda}(x;t^{\alpha},t)}{(1-t)^{|\lambda|}}.$

######  Remark 2.5.

The Jack functions $J^{(\alpha)}_{\lambda}(x)$ are the bridge to connect the
(integral) Macdonald functions with the zonal functions
($\alpha=2,O(N,\mathbb{R})$), the Schur functions ($\alpha=1,U(N,\mathbb{C})$)
and the symplectic zonal functions ($\alpha=\frac{1}{2},Sp(N,\mathbb{H})$),
[27], [28].

In our paper, we shall focus on the relationships between the integral
Macdonald functions with the power sum functions. Because the power sum
symmetric functions $\\{p_{\Delta|\Delta\in{\mathbb{P}}}\\}$ form an
${\mathbb{F}}$-basis of $\Lambda_{\mathbb{F}}$, and hence we may express the
integral Macdonald function $J_{\lambda}(x;q,t)$ in terms of them, say

(56) $J_{\lambda}(x;q,t)=\sum_{\Delta}a_{\lambda}(\Delta)(q,t)p_{\Delta},$

where $a_{\lambda}(\Delta)(q,t)\in{\mathbb{F}},$ especially, we denote
$dim\lambda=a_{\lambda}(1^{|\lambda|})(q,t),$ which is not generally the unit
and very important to construct the Macdonald-Hurwitz numbers.

######  Remark 2.6.

We believe that $dim\lambda=a_{\lambda}(1^{|\lambda|})(q,t)$ is some kind of
the normalized dimension of the irreducible representation, in fact, for the
case $q=t,$ which is actually the normalized dimension of the irreducible
complex representation symmetrical group $S_{|\lambda|}$ corresponding to the
partition $\lambda.$ For the case of the Jack functions, $dim\lambda$ is the
unit, [28], p382, (10.29).

Then we can restate the Orthogonal Lemma/Cutting Lemma as

######  Lemma 2.7.

  * (1)

The first orthogonal lemma

(57)
$\qquad\sum_{\\{\Delta||\Delta|=|\lambda|\\}}a_{\lambda}(\Delta)(q,t)z_{\Delta}(q,t)a_{\mu}(\Delta)(q,t)=\delta_{\lambda\mu}j_{\lambda}(q,t).$

  * (2)

The second orthogonal lemma

(58)
$\sum_{\\{\lambda||\lambda|=|\Delta_{1}|=|\Delta_{2}|\\}}a_{\lambda}(\Delta_{1})(q,t)\frac{1}{j_{\lambda}(q,t)}a_{\lambda}(\Delta_{2})(q,t)=\delta_{\Delta_{1},\Delta_{2}}\frac{1}{z_{\Delta_{1}}(q,t)}.$

######  Remark 2.8.

For the orthogonality of the Jack functions, please refer to [28], p382,
(10.31).

###  2.5. The initial values

In this section, we introduce an initial value of the differential equations
(LABEL:3).

If $a$ is indeterminate, we denote by $(a;q)_{\infty}$ the formal infinite
product

(59) $(a;q)_{\infty}=\prod_{i=0}^{\infty}(1-aq^{i})$

regarded as a formal power series in $a$ and $q$. Let now
$x=(x_{1},x_{2},...)$ and $y=(y_{l},y_{2}...)$ be two sequences of independent
indeterminate, we define [28], p309, (2.5),

(60)
$\Pi(x,y;q,t)=\prod_{i,j}\frac{(tx_{i}y_{j};q)_{\infty}}{(x_{i}y_{j};q)_{\infty}}.$

Then we have [28], p324, (4.13); p309, (2.6)

(61) $\displaystyle\Pi(x,y;q,t)$ $\displaystyle=$
$\displaystyle\sum_{\lambda}j_{\lambda}(q,t)^{-1}J_{\lambda}(x;q,t)J_{\lambda}(y;q,t),$
(62) $\displaystyle=$
$\displaystyle\sum_{\Delta}z_{\Delta}(q,t)^{-1}p_{\Delta}(x)p_{\Delta}(y),$

where $J_{\lambda}(x;q,t)$ and $J_{\lambda}(y;q,t)$ is the integral Macdonald
functions in variables $x$ and $y,$ respectively. In the
Lemma(LABEL:perfect0), we will give out a geometric proof for the above last
two equalities.

##  3\. Macdonald-Hurwitz number

After a long preparation, we could define our Macdonald-Hurwitz numbers in
this section.

Let $\Sigma^{h}$ be a closed connected Riemann surface of genus $h\geq 0,$ and
$\Sigma^{g}$ a closed connected Riemann surface of genus $g\geq 0$. For a
general given points set $\\{q_{1},\cdots,q_{k}\\}\in\Sigma^{h}$, which is
called the set of branch points, we call a holomorphic map
$f:\Sigma^{g}\to\Sigma^{h}$ a ramified covering of $\Sigma^{h}$ of degree
$d\geq 0$ by $\Sigma^{g}$ with a ramification type
$(\Delta_{1},\cdots,\Delta_{k})$, if the preimages of
$f^{-1}(q_{i})=\\{c_{i}^{1},\cdots,c_{i}^{l_{i}}\\}$ with orders
$\Delta_{i}=(\delta^{i}_{1},\cdots,\delta^{i}_{l_{i}})\vdash d$ for
$i=1,\cdots,k,$ respectively, where $l_{i}=l(\Delta_{i})$ is the length of the
partition $\Delta_{i}.$

######  Definition 3.1.

We define the connected genus $g\geq 0$ Macdonald-Hurwitz number with degree
$d>0$ over the closed connected Riemann surfaces $\Sigma^{h}$ with the
ramification type $(\Delta_{1},\cdots,\Delta_{k})$ in terms of the element of
the transformation matrix from the integral Macdonald functions to the power
sum as

(63) $\displaystyle\hskip
56.9055ptMH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})=\sum_{\lambda\vdash
d}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{1-h}\frac{a_{\lambda}(\Delta_{1})(q,t)}{dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{k})(q,t)}{dim\lambda},$

under the condition

(64) $(2-2h)d-(2-2g)=\sum_{i=1}^{k}(d-l(\Delta_{i})),$

otherwise, we define

(65) $\displaystyle{}MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})=0,$

and once we take $d=0,$ it is convenient to define

(66) $\displaystyle MH_{h}^{g,0}(\Delta_{1},\cdots,\Delta_{k})=0.$

To define the disconnected Macdonald-Hurwitz number, we take the rule of the
exponential-logarithmic correspondence, i.e., the disconnected Macdonald-
Hurwitz numbers are defined by the product of all connected the Macdonald-
Hurwitz numbers module their automorphisms, and then sum all possibilities up,
i.e., for any integer $g\in\mathbb{Z},$

(67) $MH^{g,d}_{h}(\Delta_{1},\cdots,\Delta_{k})_{disconnected}=\sum_{m\geq
1}\sum\frac{1}{|Aut|}\prod_{i=1}^{m}MH^{g_{i},d_{i}}_{h}(\Delta_{1}^{i_{1}},\cdots,\Delta_{k}^{i_{k}}),$

where the sum satisfies the following conditions: for a fixed component number
$m$ of the Macdonald-Hurwitz numbers, we have

(68) $\displaystyle g_{i}\geq 0,\textit{for $i=1,\cdots,m$}$ (69)
$\displaystyle d_{i}\geq 0,\textit{for $i=1,\cdots,m$}$ (70)
$\displaystyle\Delta_{1}^{i_{1}},\cdots,\Delta_{k}^{i_{k}}\vdash
d_{i}\textit{or 0, for $i=1,\cdots,m$}$ (71)
$\displaystyle(2-2h)d-(2-2g_{i})=\sum_{j=1}^{k}(d_{i}-l(\Delta_{j}^{i_{j}})),\textit{for
$i=1,\cdots,m$}$ (72)
$\displaystyle\Delta_{j}=\cup_{i=1}^{m}\\{\Delta_{j}^{i_{j}}\\},\textit{for
$j=1,\cdots,k$}$ (73) $\displaystyle 1-g=\sum_{i=1}^{m}(1-g_{i})$ (74)
$\displaystyle d=\sum_{i=1}^{m}d_{i},$

where the formula (72) means that the partitions $\Delta_{j}^{i_{j}}$ for
$i=1,\cdots,m,$ compose the partition $\Delta_{j}$, i.e., the ramification
type of the $j$-th branch point $q_{j}$ is invariant when we split the Riemann
surface $\Sigma^{g}$ into m components, ($\Delta_{j}^{i_{j}}$ may be a empty
partition, i.e, $\Delta_{j}^{i_{j}}\vdash 0,$ the $j$-th branch point $q_{j}$
might not have pre-image in the $i$-th component.), and $|Aut|$ is the
cardinality of the automorphisms of all connected components. It is easy to
check that the disconnected Macdonald-Hurwitz numbers are the exponential
functions of the connected ones. To simply our notation, we shall denote the
Macdonald-Hurwitz numbers by $MH^{g,d}_{h}(\Delta_{1},\cdots,\Delta_{k})$
neglecting the connectedness or the disconnectedness.

######  Remark 3.2.

If we take the limit along the path as $q=t^{\alpha},t\to 1$, the Macdonald
functions will be specialized to the Jack functions, and thus the Macdonald-
Hurwitz numbers can be referred to as the Jack-Hurwitz numbers. This is
similar to the $b$-Hurwitz numbers ($b=\alpha-1$) discussed in [7].

By the first orthogonal Lemma, we can iteratively calculate the Macdonald-
Hurwitz numbers by decreasing the genus of the target surface or the numbers
of the branch points.

######  Theorem 3.3.

  * (1)

Decrease the genus of the target surface directly, i.e., if $h\geq 1,$ we have

(75)
$\displaystyle{}MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})=\sum_{\Delta\vdash
d}MH_{h-1}^{g,d}(\Delta_{1},\cdots,\Delta_{k},\Delta,\Delta)z_{\Delta}(q,t),$

  * (2)

(Associativity) Decrease the genus of the target surface and the numbers of
the branched points by cutting it into two parts, i.e, for
$h_{1}+h_{2}=h,h_{1}\geq 0,h_{2}\geq 0,$ and any positive integer $k>l\geq 1$,
we have

$\displaystyle MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})$ $\displaystyle=$
$\displaystyle\sum_{\Delta}MH_{h_{1}}^{g_{1},d}(\Delta_{1},\cdots,\Delta_{l},\Delta)z_{\Delta}(q,t)$
$\displaystyle\times
MH_{h_{2}}^{g_{2},d}(\Delta,\Delta_{l+1},\cdots,\Delta_{k}).$

where the genus $g,g_{1},g_{2}$ are determined by the formula
$(\ref{hurwitz})$.

###### Proof.

  * (1)

If $h\geq 1,$ by the first orthogonal lemma, we have

$\displaystyle MH_{h}^{g,d}(\Delta_{1},\cdots,\Delta_{k})$ $\displaystyle=$
$\displaystyle\sum_{\lambda\vdash
d}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{1-h}\frac{a_{\lambda}(\Delta_{1})(q,t)}{\dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{k})(q,t)}{dim\lambda}$
(77) $\displaystyle=$ $\displaystyle\sum_{\lambda\vdash
d}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{1-h}\frac{a_{\lambda}(\Delta_{1})(q,t)}{dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{k})(q,t)}{dim\lambda}$
$\displaystyle\times\sum_{\Delta,\mu}a_{\lambda}(\Delta)(q,t)z_{\Delta}(q,t)a_{\mu}(\Delta)(q,t)\frac{\delta_{\lambda\mu}}{j_{\lambda}(q,t)}$
$\displaystyle=$ $\displaystyle\sum_{\lambda\vdash
d}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{1-(h-1)}\frac{a_{\lambda}(\Delta_{1})(q,t)}{dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{k})(q,t)}{dim\lambda}$
$\displaystyle\times\sum_{\Delta}\frac{a_{\lambda}(\Delta)(q,t)}{dim\lambda}z_{\Delta}(q,t)\frac{a_{\lambda}(\Delta)(q,t)}{dim\lambda}$
$\displaystyle=$ $\displaystyle\sum_{\Delta\vdash
d}MH_{h-1}^{g,d}(\Delta_{1},\cdots,\Delta_{k},\Delta,\Delta)z_{\Delta}(q,t).$

  * (2)

By the definition of the Macdonald-Hurwitz (63) and the first orthogonal
lemma, we have

$\displaystyle\mathrm{RHS}$ $\displaystyle=$
$\displaystyle\sum_{\Delta}\sum_{\lambda}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{1-h_{1}}\frac{a_{\lambda}(\Delta_{1})(q,t)}{dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{l})(q,t)}{dim\lambda}\frac{a_{\lambda}(\Delta)(q,t)}{dim\lambda}z_{\Delta}(q,t)$
$\displaystyle\times\sum_{\mu}(\frac{dim^{2}\mu}{j_{\mu}(q,t)})^{1-h_{2}}\frac{a_{\mu}(\Delta)(q,t)}{dim\mu}\frac{a_{\mu}(\Delta_{l+1})(q,t)}{dim\mu}\cdots\frac{a_{\mu}(\Delta_{k})(q,t)}{dim\mu}$
$\displaystyle=$
$\displaystyle\sum_{\lambda}(\frac{dim^{2}\lambda}{j_{\lambda}(q,t)})^{2-h_{1}-h_{2}}\frac{a_{\lambda}(\Delta_{1})(q,t)}{dim\lambda}\cdots\frac{a_{\lambda}(\Delta_{l})(q,t)}{dim\lambda}\delta_{\lambda\mu}\frac{j_{\lambda}(q,t)}{dim^{2}\lambda}$
$\displaystyle\times\sum_{\mu}\frac{a_{\mu}(\Delta_{l+1})(q,t)}{dim\mu}\cdots\frac{a_{\mu}(\Delta_{k})(q,t)}{dim\mu}$
$\displaystyle=$ $\displaystyle LHS.$

∎

######  Corollary 3.4.

For any positive integer $d,$ let $(d)\vdash d$ be a partition of $d,$ then we
have

(78)
$MH_{0}^{0,d}((d),(d))=\frac{1}{z_{(d)}(q,t)}=\frac{1-t^{d}}{d(1-q^{d})},$

especially, taking $d=1,$ we have

(79) $MH_{0}^{0,1}((1),(1))=\frac{1}{z_{(1)}(q,t)}=\frac{1-t}{1-q}.$

###### Proof.

It follows from the Theorem 3.3 (2). ∎

######  Remark 3.5.

If there is only one branched point in the case of degree one, we have

(80) $MH_{0}^{0,1}((1))=\frac{1-t}{1-q},$

which follows from that $J_{(1)}(x;q,t)=(1-t)P_{(1)}(x;q,t)=(1-t)p_{1}$, [28],
p323, (4.8), which does not conflict with the classical Hurwitz number,
because in the special case, we have first to deal $j_{(1)}(q,t)$ and
$a_{(1)}((1))(q,t),$ i.e.,

(81) $\displaystyle
j_{(1)}(q,t)=c_{(1)}(q,t)c^{\prime}_{(1)}(q,t)=(1-t)(1-q),$ (82)
$\displaystyle j_{(1)}(1,1)=\lim_{t\to 1}\frac{j_{(1)}(t,t)}{(1-t)^{2}}=1.$

More generally, if taking $q=t^{\alpha},$ and letting $t\to 1,$ the formula
(80) would be understood as

(83) $MH_{0}^{0,1}((1))(\alpha)=\lim_{t\to
1}\frac{1-t}{1-q}=\frac{1}{\alpha},$

where $MH_{0}^{0,1}((1))(\alpha)$ is the Jack-Hurwitz numbers with the Jack
parameter $\alpha.$

######  Example 3.6.

By the Remark (3.5), for $d\geq 1,$
$MH_{0}^{1-d,d}((1^{d}))=\frac{1}{d!}\frac{(1-t)^{d}}{(1-q)^{d}}$, which is a
disconnected Macdonald-Hurwitz number (with $m=d$ same components).

######  Remark 3.7.

It is obvious that $a_{\lambda}(\Delta)(q,t)=0$ unless $|\Delta|=|\lambda|$
[28], p356. If $|\Delta|\textgreater|\lambda|,$ we could take
$a_{\lambda}(\Delta)(q,t)=0$, however, if $|\Delta|\textless|\lambda|,$ we
could shift the weight of $\Delta$ by adding $|\lambda|-|\Delta|$-numbers 1,
i.e., we get a new partition
$\Delta^{\prime}=1^{m_{1}(\Delta)+(|\lambda|-|\Delta|)}2^{m_{2}(\Delta)}\cdots$,
if $\Delta=1^{m_{1}(\Delta)}2^{m_{2}(\Delta)}\cdots$, such that
$|\Delta^{\prime}|=|\lambda|$, then we could define
$a_{\lambda}(\Delta)(q,t):={{m_{1}(\Delta)}\choose{|\lambda|-|\Delta|+m_{1}(\Delta)}}a_{\lambda}(\Delta^{\prime})(q,t)$,
[30],[40], then we would obtain a shift Macdonald-Hurwitz number, but in this
paper, we shall always assume $|\Delta|=|\lambda|$.

######  Remark 3.8.

  * (1)

The equation (64) is exact the Hurwitz formulas.

  * (2)

Taking $q=t$, and letting $t\to 1$, we have

$\displaystyle J^{(1)}_{\lambda}(x)$ (84) $\displaystyle=\lim_{t\to
1}\frac{J_{\lambda}(x;t,t)}{(1-t)^{|\lambda|}}$ $\displaystyle=\lim_{t\to
1}\frac{s_{\lambda}\prod_{s\in\lambda}(1-t^{hook(s)})}{(1-t)^{|\lambda|}}$
$\displaystyle=s_{\lambda}\prod_{s\in\lambda}hook(s)$
$\displaystyle=\sum_{\Delta}\left[\frac{|\lambda|!}{dim(\lambda)}\frac{\chi_{\lambda(\Delta)}}{z_{\Delta}}\right]p_{\Delta},$

where $s_{\lambda}$ is the Schur function, and $dim(\lambda),$
$\chi_{\lambda}(\Delta)$ are the dimensions and the character of the
irreducible complex representation of symmetric group $S_{d}$ associated with
$\lambda$ and $\Delta,$ respectively, [28], [30], [39]. Thus we have

(85) $\displaystyle j_{\lambda}(1,1)=(\frac{|\lambda|!}{dim(\lambda)})^{2},$
(86) $\displaystyle
a_{\lambda}(\Delta)(1,1)=\frac{|\lambda|!}{dim(\lambda)}\frac{\chi_{\lambda}(\Delta)}{z_{\Delta}},$
(87) $\displaystyle
a_{\lambda}(1^{|\lambda|})(1,1)=\frac{|\lambda|!}{dim(\lambda)}\frac{\chi_{\lambda}(1^{|\lambda|})}{z_{(1^{|\lambda|})}}=1.$

Substitute the above data (85)-(87) into (63), we return to the classical
Hurwitz number, [34], [39], p4, (8), it is an exact reason to make us choose
the integral Macdonald functions with two parameters as our objects, rather
than the Macdonald functions. Moreover, we could obtain that, under the
condition (64), the Macdonald-Hurwitz numbers are nonzero, since the classical
Hurwitz numbers are nonzero due to the Riemannian existence theorem. The other
reason for us to choose the integral Macdonald functions is that the integral
Macdonald functions have better integrality than the Macdonald functions, for
example, the normal of the integral Macdonald functions are the polynomial
functions in the parameters $q,t,$ however, the normal of the Macdonald
functions are the rational functions in the parameters $q,t,$ i.e., [28], p323
(4.11), p339 (6.19), p353. (8.7))

(88) $\displaystyle<J_{\lambda}(x;q,t),J_{\lambda}(x;q,t)>_{q,t}$
$\displaystyle=$ $\displaystyle c_{\lambda}(q,t)c^{\prime}_{\lambda}(q,t)$
(89) $\displaystyle=$
$\displaystyle\prod_{s\in\lambda}(1-q^{a(s)}t^{l(s)+1})(1-q^{a(s)+1}t^{l(s)}),$
$\displaystyle<P_{\lambda}(x;q,t),P_{\lambda}(x;q,t)>_{q,t}$ $\displaystyle=$
$\displaystyle\frac{c^{\prime}_{\lambda}(q,t)}{c_{\lambda}(q,t)}$
$\displaystyle=$
$\displaystyle\prod_{s\in\lambda}\frac{(1-q^{a(s)+1}t^{l(s)})}{(1-q^{a(s)}t^{l(s)+1})}.$

  * (3)

Due to the theorem (3.3)(1), we mainly consider the special case $h=0$ in the
rest of the paper, and denote the Macdonald-Hurwitz numbers
$MH_{0}^{g,d}(\Delta_{1},\cdots,\Delta_{k})$ by
$MH^{g}_{d}(\Delta_{1},\cdots,\Delta_{k}).$

##  4\. The genus-expanded cut-and-join operators/ integral Macdonald
functions with two parameters $(q,t)$

In this section, we assume all partitions are the partitions of $d.$ Let
$\hbar$ be indeterminate, then we have

######  Definition 4.1.

Following the idea of [39], for any partition $\Delta\vdash d$, we define a
genus-expanded cut-and-join operator $D(\Delta,\hbar)$ by

(90) $D(\Delta,\hbar)=\sum_{\Gamma,\Gamma^{\prime}\vdash
d}z_{\Gamma^{\prime}}(q,t)\hbar^{d+l(\Gamma^{\prime})-l(\Delta)-l(\Gamma)}MH_{d}^{g}(\Gamma^{\prime},\Delta,\Gamma)p_{\Gamma}\frac{\partial}{\partial
p_{\Gamma^{\prime}}},$

where genus $g$ satisfies the formula (64) ($h=0$):

(91) $2d-(2-2g)=(d-l(\Gamma^{\prime}))+(d-l(\Delta))+(d-l(\Gamma)),$

and $\frac{\partial}{\partial p_{\Gamma^{\prime}}}$ is defined by the formula
(38).

######  Remark 4.2.

  * (1)

Taking the connected Macdonald-Hurwitz number in the above operator (90), and
$\Delta=(1^{d-2}2)$ and $q=t,$ letting $t\to 1,$ we return the classical cut-
and-join operator [10], [11], [12], [22], [30] or the Laplace-Beltrami
operator (8) up a multiple $\frac{1}{2},$ thus the above operator (90) can be
thought as a generalization of the Laplace-Beltrami operator (8).

  * (2)

As in [39], the power $d+l(\Gamma^{\prime})-l(\Delta)-l(\Gamma)$ of $\hbar$ in
the formula (90) is the "lost" genus after we have cut the target Riemann
surface $\Sigma^{h}$ into two parts, referring to the Remark(5.8).

In the rest of this section, we establish a connection between the new
commutative and associative algebra $(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ and
the middle-dimensional $\mathbb{C}^{*}$-equivalent cohomological ring via the
Jack function over the Hilbert scheme points of $\mathbb{C}^{2}$ constructed
by W. Li, Z. Qin, and W. Wang [26].

Let $\alpha$ be the Jack parameter of the Jack functions
$J^{(\alpha)}_{\lambda}(x)$, and $\hat{\mathbb{F}}:=\mathbb{Q}(A,B)$ be the
field of the rational functions with the rational coefficient in $A,B,$ where
$A,B$ are two indeterminates such that $\alpha=\frac{B}{A}$.

Let $C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})$ be the restriction of
$(C(\mathbb{F}[S_{d}]),\circ_{q,t})$ to the special path $\eta(A|B)$ defined
by

(140) $\displaystyle t=r^{A},$ (141) $\displaystyle q=r^{B},$

where $r$ is the path parameter, and letting $r\to 1$, which is different from
the path $q=t^{\alpha},t=t\to 1$ in the Macdonald’s book [28] or the formula
(55), although we will eventually obtain the same Jack functions with the Jack
parameters $\alpha=\frac{B}{A}$ from the Macdonald functions in both case. We
denote the corresponding quantities by adding $A|B$, for example, we have the
following data:

(142) $\displaystyle c_{\lambda}(A|B):=\lim_{r\to
1}\frac{c_{\lambda}(r^{B},r^{A})}{(1-r)^{|\lambda|}}=\prod_{(i,j)\in\lambda}(A(l(i,j)+1)+Ba(i,j)),$
(143) $\displaystyle c^{\prime}_{\lambda}(A|B):=\lim_{r\to
1}\frac{c^{\prime}_{\lambda}(r^{B},r^{A})}{(1-r)^{|\lambda|}}=\prod_{(i,j)\in\lambda}(Al(i,j)+B(a(i,j)+1)),$
(144) $\displaystyle j_{\lambda}(A|B):=\lim_{r\to
1}\frac{j_{\lambda}(r^{B},r^{A})}{(1-r)^{2|\lambda|}}=c_{\lambda}(A|B)c_{\lambda}^{\prime}(A|B).$

We should notice that the formula (142)-(144) is different from the formula
(49)-(51) up to a factor $A^{|\lambda|}$ or $A^{2|\lambda|}$ respectively,
since we have taken the limit along the different paths. Moreover the above
descriptions are the exact means of "we take the limit of some qualities along
a path", or "we take the restrict of a ring to a path".

Moreover we have the bilinear form $<-,->_{A|B}$ and trilinear form
$T_{A|B}(-,-,-)$ over $C(\hat{\mathbb{F}}[S_{d}])$ as

(145)
$\displaystyle<C_{\Delta_{1}},C_{\Delta_{2}}>_{A|B}=\delta_{\Delta_{1},\Delta_{2}}\frac{1}{z_{\Delta_{1}}\alpha^{l(\Delta_{1})}};$
(146) $\displaystyle T_{A|B}(C_{\Delta_{1}},C_{\Delta_{2}},C_{\Delta_{3}})$
(147)
$\displaystyle:=<C_{\Delta_{1}},C_{\Delta_{2}}\circ_{A|B}C_{\Delta_{3}}>_{A|B}$
(148)
$\displaystyle=<C_{\Delta_{1}}\circ_{A|B}C_{\Delta_{2}},C_{\Delta_{3}}>_{A|B}$
(149) $\displaystyle=MH_{d}^{g}(\Delta_{1},\Delta_{2},\Delta_{3})(A|B),$

where $MH_{d}^{g}(\Delta_{1},\Delta_{2},\Delta_{3})(A|B)$ is the limit of the
Macdonald-Hurwitz numbers $MH_{d}^{g}(\Delta_{1},\Delta_{2},\Delta_{3})$ along
the path $\eta(A|B)$.

We restate the Corollary (LABEL:6.1) as

######  Lemma 6.5.

$C(\hat{\mathbb{F}}[S_{d}])$ is a Frobenius algebra with the unit
$C_{(1^{d})}$ for the bilinear form $<-,->_{A|B}$ and trilinear form
$T_{A|B}(-,-,-)$. Moreover, for $\lambda\vdash d,$

(150) $\displaystyle\epsilon_{\lambda}(A|B):=\lim_{r\to
1}\epsilon_{\lambda}(r^{B},r^{A})(1-r)^{|\lambda|}$

are the idempotent bases of $(C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})$, which
are orthogonal to each other,

(151)
$\displaystyle<\epsilon_{\lambda}(A|B),\epsilon_{\mu}(A|B)>_{A|B}=\delta_{\lambda,\mu}\frac{1}{j_{\lambda}(A|B)}.$

###### Proof.

It follows from the Corollary(LABEL:6.1) and $dim\lambda=1$ for the Jack
functions, [28], p382, (10.29). ∎

In the rest of the paper, we will adopt the same notations as [25], [26].

Let $\theta$ be the 1-dimensional standard
$\mathbb{T}:=\mathbb{C}^{\star}$-module. Fix two nonzero integers $A$ and $B$
with the same signs, which are corresponding to the parameters $\alpha,\beta$
in [25], [26]. Let $u,v$ be the standard coordinate functions on
$\mathbb{C}^{2}$. For $s\in\mathbb{T}$, we define the action of $\mathbb{T}$
on $\mathbb{C}^{2}$ by [26], Example 2.1,

(152) $s\cdot(u,v)=(s^{A}u,s^{-B}v).$

The origin of $\mathbb{C}^{2}$ is the only fixed point, which is denoted by
$O$.

Let $H^{\star}_{\mathbb{T}}(\mathbb{C}^{2})$ be the equivariant cohomology
ring of $\mathbb{C}^{2}$ with $\mathbb{C}$ coefficients.
$H^{\star}_{\mathbb{T}}(\mathbb{C}^{2})$ is a $\mathbb{C}[\bar{t}]$-module if
we identify $H^{\star}_{\mathbb{T}}(pt)$ with $\mathbb{C}[\bar{t}]$ ($\bar{t}$
is an element of the degree -2).

Let $F$ and $F^{\prime}$ be the $u$-axis and $v$-axis respectively in
$\mathbb{C}^{2}$. As $\mathbb{T}$-modules, we have $T_{O}F=\theta^{-A}$ and
$T_{O}F^{\prime}=\theta^{B}$, then we have [26]

(153)
$\displaystyle[F]=-A^{-1}\bar{t}^{-1}[O],[F^{\prime}]=B^{-1}\bar{t}^{-1}[O],$

Let $(\mathbb{C}^{2})^{[d]}$ be the Hilbert scheme parameterizing all the
0-dimensional closed subschemes $\xi$ of $\mathbb{C}^{2}$ with
$dim_{\mathbb{C}}H^{0}(\mathcal{O}_{\xi})=d$. The $\mathbb{T}$\- action on
$\mathbb{C}^{2}$ induces a $\mathbb{T}$-action on $(\mathbb{C}^{2})^{[d]}$.
The support of a $\mathbb{T}$-fixed point in $(\mathbb{C}^{2})^{[d]}$ is
contained in $(\mathbb{C}^{2})^{\mathbb{T}}=O$ and indexed by partitions
$\lambda$ of $d$.

Let $\xi_{\lambda}$ be the fixed points in
$((\mathbb{C}^{2})^{[d]})^{\mathbb{T}}$ corresponding to a partition $\lambda$
of $d$. Then for $\lambda\vdash d$, the tangent space of
$(\mathbb{C}^{2})^{[d]}$ at the fixed point $\xi_{\lambda}$ is
$\mathbb{T}$-equivariantly isomorphic to [26], p4, (2.10):

(154)
$T_{\xi_{\lambda}}(\mathbb{C}^{2})^{[d]}=\oplus_{(i,j)\in\lambda}\theta^{A(l(i,j)+1)+Ba(i,j)}\oplus\theta^{-Al(i,j)-B(a(i,j)+1)}.$

We have the equivariant Euler class [26], p4, (2.11),

(155)
$e_{\mathbb{T}}(T_{\xi_{\lambda}}(\mathbb{C}^{2})^{[d]})=(-1)^{d}c_{\lambda}(A|B)c^{\prime}_{\lambda}(A|B)\bar{t}^{2d}.$

Note that $\xi_{\lambda}\in H^{4d}_{\mathbb{T}}((\mathbb{C}^{2})^{[d]})$. Then
for $\lambda\vdash d,$ the following distinguished class [26], p4, (2.14):

(156)
$[\lambda]=(-1)^{d}\frac{1}{c_{\lambda}(A|B)}\bar{t}^{-d}[\xi_{\lambda}]$

form a linear basis of the $\mathbb{C}$-vector space
$H^{2d}_{\mathbb{T}}((\mathbb{C}^{2})^{[d]})$, which have $\star$-product
structure [26], p8,

(157)
$\frac{[\lambda]}{c^{\prime}_{\lambda}(A|B)}\star\frac{[\mu]}{c^{\prime}_{\mu}(A|B)}=\delta_{\lambda,\mu}\frac{[\lambda]}{c^{\prime}_{\lambda}(A|B)},\lambda,\mu\vdash
d,$

for $[\lambda],[\mu]\in H^{2d}_{\mathbb{T}}((\mathbb{C}^{2})^{[d]})$ and the
bilinear form $<-,->$ [26], p8,

(158)
$<[\lambda],[\mu]>=\delta_{\lambda,\mu}\frac{c^{\prime}_{\lambda}(A|B)}{c_{\lambda}(A|B)}.$

We have the following theorem,

######  Theorem 6.6.

There exists an isomorphism that preserves the bilinear form between
$(C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})$ and
$(H^{2d}_{\mathbb{T}}((\mathbb{C}^{2})^{[d]}),\star)$.

###### Proof.

Applying the lemma (6.5), let us construct the isomorphism which maps the
idempotent element to the idempotent element, i.e., for $\lambda\vdash d,$

(160) $\displaystyle\rho:$
$\displaystyle(C(\hat{\mathbb{F}}[S_{d}]),\circ_{A|B})\to(H^{2d}_{\mathbb{T}}((\mathbb{C}^{2})^{[d]}),\star)$
$\displaystyle\epsilon_{\lambda}(A|B)\mapsto\frac{[\lambda]}{c^{\prime}_{\lambda}(A|B)}.$

Moreover, we have

(161) $\displaystyle<\rho(\epsilon_{\lambda}(A|B)),\rho(\epsilon_{\mu}(A|B))>$
(162)
$\displaystyle=<\frac{[\lambda]}{c^{\prime}_{\lambda}(A|B)},\frac{[\mu]}{c^{\prime}_{\mu}(A|B)}>$
(163) $\displaystyle=\delta_{\lambda,\mu}\frac{1}{j_{\lambda}(A|B)}$ (164)
$\displaystyle=<\epsilon_{\lambda}(A|B),\epsilon_{\mu}(A|B)>_{A|B}.$

Thus $\rho$ is the isomorphism we need. ∎

######  Remark 6.7.

The equivariant cohomological class $[\lambda]$ can be corresponded to the
Jack polynomial, as detailed in Theorem 3.2 of [26].

######  Remark 6.8.

In our views, $(H^{2d}_{\mathbb{T}}(\mathbb{C}^{2})^{[d]},\star)$ constructed
by W. Li, Z. Qin, and W. Wang should be called $\mathbb{T}$-equivalent
cohomological ring via the special path $q=r^{B},t=r^{A},r\to 1$ over the
Hilbert scheme points of $\mathbb{C}^{2}$

(with two parameters $A,B$), rather than via the Jack functions ( which should
take the limit along the path $q=t^{\alpha},t\to 1,$ with only one parameter
$\alpha,$ in fact, W. Li, Z. Qin, and W. Wang actually took the limit along
the path $\eta(A|B)$ in [26]), whose idempotent elements in both cases are
different up to a factor $A^{|\lambda|}$.

Acknowledgements The author would like to thank Prof. Xiaojun Chen, Prof. An-
min Li, Prof. Wei Luo and Prof. Yongbin Ruan, Prof. Qi Zhang, Guanghui Fu, and
Yin Liu for their communications and helps. This work was partially supported
by NSFC 11890663 and NSFC 12171351, which are led by Prof. Bohui Chen and
Prof. Song Yang, respectively.

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