March 12, 2026
In this paper, based on the value of central character on the transposition, we find structure and large genus asymptotics of certain Hurwitz numbers.
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The notion of Hurwitz numbers was introduced in [1], [2]. The question is to count the weighted number \(H^*_d(\theta^{(1)}, \dots , \theta^{(n)})\) of ramified coverings of degree \(d\) of the Riemann sphere \(\mathbb{P}^1\) with the ramification profiles \(\theta^{(1)},\dots,\theta^{(n)}\vdash d\). Here, \(\theta\vdash d\) denotes a partition \(\theta\) of weight \(d\). The ramified covering is called connected if the upper Riemann surface is connected. The weighted number of connected ramified coverings of genus \(g\) and degree \(d\) with the ramification profiles \(\theta^{(1)},\dots,\theta^{(n)}\vdash d\) is called connected Hurwitz numbers (for short Hurwitz numbers), denoted by \(H_{g,d}(\theta^{(1)}, \dots , \theta^{(n)})\).
For a partition \(\theta=(\theta_1,\dots,\theta_{l})\vdash d\) with \(\theta_1\geq\theta_2\geq\cdots\geq\theta_{l}> 0\), denote \(|\theta|=\sum_{i=1}^{l}\theta_{i}=d,l(\theta)=l\), \(l^*(\theta)=d-l\) and \(z_\theta=\prod\limits_i m_i(\theta)! i^{m_i}\) with \(m_i(\theta)\) being the multiplicity of \(i\) in \(\theta\).
In this paper, we will prove
Theorem 1. For any fixed \(d\geq5,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), we have \[\begin{align} &H_{g,d}(\mu^{(1)},\dots,\mu^{(s)},2\,1^{d-2},2\,1^{d-2},\dots)\nonumber\\ =&\frac{2}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \sum_{1\leq m\leq \tbinom{d}{2}}b(\mu^{(1)},\dots,\mu^{(s)},m) m^{2g+2d-\sum_{i=1}^s l^*(\mu^{(i)})-2}, \label{cHv} \end{align}\qquad{(1)}\] where \(b(\mu^{(1)},\dots,\mu^{(s)},m)\) are rational numbers satisfying
1. \(b(\mu^{(1)},\dots,\mu^{(s)},\tbinom{d}{2})=1\);
2. \(b(\mu^{(1)},\dots,\mu^{(s)},m)=0\), for \(\tbinom{d-1}{2}<m<\tbinom{d}{2}\);
3. \(b(\mu^{(1)},\dots,\mu^{(s)},\tbinom{d-1}{2})=-d^{2-s}\prod_{i=1}^{s}m_1(\mu^{(i)})\);
4. \(b(\mu^{(1)},\dots,\mu^{(s)},\frac{d(d-3)}{2})=(d-1)^{2-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\).
For the case \(s=0\), Theorem 1 was found and proved by Hurwitz [1] using representation of the symmetric group. This proof was briefly reviewed in [3]. Our proof of Theorem 1 for general \(s\) is along this line. For the case \(s=1\), Do-He-Robertson [4] proved the statement 1 of Theorem 1, and conjectured the statement 2 which was later proved by Yang [5]. For the case \(s=2\), Do-He-Robertson [4] also deduced the statement 1 with \(\forall\,\mu^{(1)}\vdash d ,\mu^{(2)}=(k^\frac{d}{k})\vdash d\) for \(k>0\). In [6], we generalize the above theorem to \(H_{g,d}(\mu^{(1)},\dots,\mu^{(s)},r\,1^{d-r},r\,1^{d-r},\dots)\) and to the situation when the genus of the underlying Riemann surface could be bigger than zero. Some other general results are also obtained in [7].
For the case \(s=0\), Dubrovin-Yang-Zagier [3] also gave new recursions for the Hurwitz numbers by simplifying the so-called Pandharipande equation, and in this way gave a new proof of the statements 1, 2 of the above theorem. For the cases \(s=1,2\), by using the method of [3], we also achieve in [8] another proof of part of the above theorem.
The following corollary easily follows from Theorem 1.
Corollary 1. For any fixed \(d\geq5,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), the asymptotics of \(H_{g,d}(\mu^{(1)},\dots,\mu^{(s)},2\,1^{d-2},2\,1^{d-2},\dots)\) is given by \[\begin{align} H_{g,d}(\mu^{(1)},\dots,&\mu^{(s)},2\,1^{d-2},2\,1^{d-2},\dots) \sim\frac{2\cdot d!^{s-2}}{z_{\mu^{(1)}} \cdots z_{\mu^{(s)}}}\Big(\binom{d}{2}^{2g+2d-\sum_{i=1}^s l^*(\mu^{(i)})-2}\nonumber\\ &-d^{2-s}\prod_{i=1}^{s}m_1(\mu^{(i)})\binom{d-1}{2}^{2g+2d-\sum_{i=1}^s l^*(\mu^{(i)})-2}\nonumber\\ &+(d-1)^{2-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)(\frac{d(d-3)}{2})^{2g+2d-\sum_{i=1}^s l^*(\mu^{(i)})-2}\nonumber\\ &+o((\frac{d(d-3)}{2})^{2g+2d-\sum_{i=1}^s l^*(\mu^{(i)})-2}) \Big),\,\,g\rightarrow \infty. \end{align}\]
In this section, we prove Theorem 1.
Proof of Theorem 1. It is known that \(H_d^{*}(\theta^{(1)}, \dots , \theta^{(n)})\) has the following formula [9], [10]: \[\label{Hurwitz1} H_d^{*}(\theta^{(1)}, \dots , \theta^{(n)}) = \sum_{ \lambda\vdash d}(\frac{\dim \lambda}{d!})^{2} \prod_{i=1}^n f_{\theta^{(i)}}(\lambda)\tag{1}\] where \(\text{dim}\lambda\) is the dimension of the irreducible representations of the symmetric group \(S(d)\) corresponding to \(\lambda\), and \[\begin{align} f_{\theta^{(i)}}(\lambda):=\frac{d!}{z_{\theta^{(i)}}}\frac{\chi_\lambda(\theta^{(i)})}{\text{dim}\lambda}.\label{f} \end{align}\tag{2}\] is the central character of \(\mu^{(i)}\). Here \(\chi_\lambda(\mu^{(i)})\) is the value of the irreducible character \(\chi_\lambda\) on the conjugacy class \(\mu^{(i)}\). According to 1 , \[\begin{align} H^{*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{2\,1^{d-2},2\,1^{d-2},\dots}_{k})=\sum_{\lambda\vdash d}\big(\frac{d!}{\dim\lambda}\big)^{s-2}(f_{(2,\,1^{d-2})}(\lambda))^k\prod_{i=1}^s \frac{\chi_\lambda(\mu^{(i)})}{z_{\mu^{(i)}}} .\label{5} \end{align}\tag{3}\] A classical result [10] of the character ratio on the transposition is \[\begin{align} \frac{\chi_\lambda(2,\,1^{d-2})}{\chi_\lambda(1^d)}=\frac{1}{\binom{d}{2}}\Big(\sum_i\binom{\lambda_i}{2}-\sum_i\binom{\lambda'_i}{2}\Big).\label{tran} \end{align}\tag{4}\] By 2 , 4 and \(\dim\lambda=\chi_\lambda(1^d)\), it is clear that \[\begin{align} f_{(2,\,1^{d-2})}(\lambda)=\sum_i\binom{\lambda_i}{2}-\sum_i\binom{\lambda'_i}{2}.\label{f2} \end{align}\tag{5}\] By 5 , \[\begin{align} |f_{(2,\,1^{d-2})}(\lambda)|<\frac{d(d-3)}{2} \end{align}\] unless \[\begin{align} &f_{(2,\,1^{d-2})}(d)=\binom{d}{2},\qquad\qquad\qquad\,\,\,\, f_{(2,\,1^{d-2})}(1^d)=-\binom{d}{2},\tag{6}\\ &f_{(2,\,1^{d-2})}(d-1,1)=\frac{d(d-3)}{2},\,\qquad f_{(2,\,1^{d-2})}(2,1^{d-2})=-\frac{d(d-3)}{2}.\tag{7} \end{align}\] According to Murnaghan-Nakayama rule [11], [12], we have \[\begin{align} &\chi_{(d)}(\mu)=1,\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\, \chi_{(1^d)}(\mu)=(-1)^{l^*(\mu)},\tag{8}\\ &\chi_{(d-1,1)}(\mu)=m_1(\mu)-1,\qquad \chi_{(2,\,1^{d-2})}(\mu)=(m_1(\mu)-1)\cdot (-1)^{l^*(\mu)}.\tag{9} \end{align}\]
Lemma 1. For any fixed \(d\geq5,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), when \(k+\sum_{i=1}^s l^*(\mu^{(i)})=\text{even}\), \[\begin{align} H^{*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{2\,1^{d-2},2\,1^{d-2},\dots}_{k})= \frac{2}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} &\sum_{1\leq m\leq \tbinom{d}{2}}b^*(\mu^{(1)},\dots,\mu^{(s)},m) m^{k}, \label{dHv2} \end{align}\qquad{(2)}\] where \(b^*(\mu^{(1)},\dots,\mu^{(s)},m)\) are rational numbers with
1. \(b^*(\mu^{(1)},\dots,\mu^{(s)},\tbinom{d}{2})=1\);
2. \(b^*(\mu^{(1)},\dots,\mu^{(s)},m)=0\), for \(\frac{d(d-3)}{2}<m<\tbinom{d}{2}\);
3. \(b^*(\mu^{(1)},\dots,\mu^{(s)},\frac{d(d-3)}{2})=(d-1)^{2-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\).
Proof. By 3 , 5 and 6 , we have the structure of disconnected Hurwitz numbers as follows: \[\begin{align} H^{*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{2\,1^{d-2},2\,1^{d-2},\dots}_{k})=\frac{2}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} &\sum_{1\leq m\leq \tbinom{d}{2}}b^*(\mu^{(1)},\dots,\mu^{(s)},m) m^{k} ,\nonumber \end{align}\] where \[\begin{align} b^*(\mu^{(1)},\dots,\mu^{(s)},m)=\frac{1}{2}\sum_{\substack{\lambda\vdash d\\|f_{(2,\,1^{d-2})}(\lambda)|=m}}\dim(\lambda)^2\big(\operatorname{sgn}(f_{(2,\,1^{d-2})}(\lambda))\big)^k\prod_{i=1}^{s}\frac{\chi_\lambda(\mu^{(i)})}{\dim(\lambda)}. \label{top1} \end{align}\tag{10}\] Since \(k+\sum_{i=1}^s l^*(\mu^{(i)})=\text{even}\), by 6 and 8 , \(b^*(\mu^{(1)},\dots,\mu^{(s)},\tbinom{d}{2})=1\). From 7 and 9 , the second large term of \(m\) in ?? is \(\frac{d(d-3)}{2}\) whose coefficients \(b^*(\mu^{(1)},\dots,\mu^{(s)},\frac{d(d-3)}{2})\) equals to \((d-1)^{2-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\). ◻
By the relationship between connected and disconnected Hurwitz numbers, we have \[\begin{align} \sum_{g,d}&\sum_{\mu^{(1)},\dots,\mu^{(s)}\vdash d}\frac{1}{q!} H_{g,d}(\mu^{(1)},\dots,\mu^{(s)},2\,1^{d-2},2\,1^{d-2},\dots)x^{q} \prod_{i=1}^s p^i_{\mu^{(i)}}\nonumber\\ =&\log(\sum_{k,d}\sum_{\mu^{(1)},\dots,\mu^{(s)}\vdash d}\frac{1}{k!} H^{*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{2\,1^{d-2},2\,1^{d-2},\dots}_{k})x^k \prod_{i=1}^s p^i_{\mu^{(i)}})\label{rlt2} \end{align}\tag{11}\] with \(q=2g+2d-\sum_{i=1}^s l^*(\mu^{(i)})-2\). Expanding the right hand side of 11 by Taylor series and taking the coefficients of \(\prod_{i=1}^s p^i_{\mu^{(i)}} x^q\) on both side, we get \[\begin{align} H_{g,d}(\mu^{(1)},&\dots,\mu^{(s)},2\,1^{d-2},2\,1^{d-2},\dots)=H^*_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{2\,1^{d-2},2\,1^{d-2},\dots}_{q})\nonumber\\ &-\frac{1}{2}\sum_{\substack{d_1,d_2\geq1\\d_1+d_2=d\\k_1,k_2\geq0\\k_1+k_2=q}}\sum_{\substack{\nu^{(1)},\dots,\nu^{(s)}\vdash d_1\\\sigma^{(1)},\dots,\sigma^{(s)}\vdash d_2\\\nu^{(i)}\cup\sigma^{(i)}=\mu^{(i)}\\i\in\{1,2,\dots,s\}}}\frac{q!}{k_1!k_2!}H^*_{d_1}(\nu^{(1)},\dots,\nu^{(s)},\underbrace{2\,1^{d_1-2},2\,1^{d_1-2},\dots}_{k_1})\nonumber\\ &\qquad\qquad\qquad\qquad\qquad \times H^*_{d_2}(\sigma^{(1)},\dots,\sigma^{(s)},\underbrace{2\,1^{d_2-2},2\,1^{d_2-2},\dots}_{k_2})+\cdots.\nonumber \end{align}\] Here, the \(\cdots\) term does not contain any \(m^q\) with \(m\geq \binom{d-1}{2}\). Notice that for \(d_1,d_2\geq 1\) such that \(d_1+d_2=d\), it holds that \[\begin{align} \binom{d_1}{2}+\binom{d_2}{2}\leq\binom{d-1}{2}.\nonumber \end{align}\] Using the binomial theorem, Theorem 1 follows from Lemma 1. ◻
I would like to thank Di Yang for his advice. This work was supported by NSFC No. 12371254 and CAS No. YSBR-032.
Email: lxiang1993@ustc.edu.cn.↩︎