Upper bound of some character ratios and large genus asymptotic behavior of Hurwitz numbers
March 12, 2026
Abstract
In [1] we found the large genus asymptotics of Hurwitz numbers for the Riemann sphere with a fixed number of general profiles and some \((2,1^{d-2})\) profiles. In this paper, motivated from [2], we generalize these results to Hurwitz numbers of an arbitrary compact Riemann surface with a fixed number of general profiles and some \((r,1^{d-r})\) profiles.
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1 Introduction↩︎
The notion of Hurwitz numbers was introduced in [3], [4]. The connected Hurtwitz numbers (for short Hurwitz numbers), denoted by \(H^X_{g,d}(\theta^{(1)}, \dots , \theta^{(n)})\), are the number of connected ramified coverings \(f: C\rightarrow X\) of degree \(d\), where \(C\) is a Riemann surface of genus \(g\), \(X\) is a Riemann surface of genus \(g(X)\), and the ramification profiles over \(n\) marked points are given by the partitions \(\theta^{(1)},\dots,\theta^{(n)}\vdash d\).
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 \(\theta\).
Hurwitz [3], [4] gave a closed formula for \(H_{g,d}(2\,1^{d-2},2\,1^{d-2},\dots)=:H_{g,d}\) with fixed \(d\), which implies their structure with fixed \(d\) and their large genus asymptotics; these were briefly reviewed in [5].
In [5] Dubrovin-Yang-Zagier obtained a simple recursion for \(H_{g,d}\) based on the Pandharipande equation, and used it to give a new proof of the structure of these numbers with fixed \(d\) and the large genus asymptotics. We [6] generalized this to double Hurwitz numbers (i.e., \(H_{g,d}(\mu^{(1)},\mu^{(2)},2\,1^{d-2},2\,1^{d-2},\dots)\)) by deriving Pandharipande-type equations.
Motivated by the work of Ding-Li-Liu-Yan [2] (cf. [7]), we are towards extending the above-mentioned results by replacing \(2\,1^{d-2}\) by \(r\,1^{d-r}\) (and by more general \(\nu\)) in this paper. This leads to the problem of finding the irreducible representations that yield the largest and second-largest character ratios for a fixed conjugacy class \(\mu\). The largest character ratio is already known [8] (see Theorem B below). For the case \(\mu=(r,\,1^{d-r})\), Frumkin-James-Roichman provided a combinatorial interpretation of the central character \(f_{(r,\,1^{d-r})}(\lambda)\) (see 1 for the definition) as a signed count of Young trees of order \(r\) contained in the diagram \(\lambda\) [9].
We will prove in Section 2 the following
Theorem 1. For any fixed \(d\geq7,2\leq r\leq d-2,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), we have \[\begin{align} &H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},r\,1^{d-r},r\,1^{d-r},\dots)\nonumber\\ =&\frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \sum_{1\leq m\leq \frac{d!}{r(d-r)!}}b^X_r(\mu^{(1)},\dots,\mu^{(s)},m) m^{\frac{1}{r-1}(2g+(2-2g(X))d-2-\sum_{i=1}^s l^*(\mu^{(i)}))}, \label{cHv} \end{align}\qquad{(1)}\] where \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_r^{X}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers with
1. \(b^X_r(\mu^{(1)},\dots,\mu^{(s)},\frac{d!}{r(d-r)!})=1\);
2. \(b^X_r(\mu^{(1)},\,\cdots,\,\mu^{(s)},m)=0\) for \(\frac{(d-1)!}{r\cdot(d-r-1)!}<m<\frac{d!}{r(d-r)!}\);
3. \(b^X_r(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{(d-1)!}{r\cdot(d-r-1)!})=-d^{2-2g(X)-s}\prod_{i=1}^{s}m_1(\mu^{(i)})\);
4. \(b^X_r(\mu^{(1)},\dots,\mu^{(s)},m)=0\) for \(\frac{(d-r-1)d!}{r(d-1)(d-r)!}<m<\frac{(d-1)!}{r\cdot(d-r-1)!}\);
5. \(b_{r}^{X}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{(d-r-1)d!}{r(d-1)(d-r)!})=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\).
For the case \(g(X)=0\), \(s=0\) and \(r=2\), Theorem 1 can be deduced from [3], [4] (cf. also [5]). For the case \(g(X)=0\), \(s=1\) and \(r=2\), Do-He-Robertson [10] proved the statement 1 in Theorem 1, and conjectured the statement 2 which was later proved by Yang [11]. Do-He-Robertson [10] also deduced the statement 1 for \(H_{g,d}(\mu^{(1)},k^\frac{d}{k},2\,1^{d-2},2\,1^{d-2},\dots)\). Some general results are also obtained in [12]. We also refer to [2], [7] for other interesting results.
Furthermore, we generalize the statement 1 of Theorem 1 to Theorem 2, by replacing \(r\,1^{d-r}\) with \(\nu\).
Theorem 2. For any fixed \(d\geq5,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)},\nu\vdash d\), we have \[\begin{align} &H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},\nu,\nu,\dots)\nonumber\\ =&\frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \sum_{1\leq m\leq \frac{d!}{z_\nu}}b^X_\nu(\mu^{(1)},\dots,\mu^{(s)},m) m^{(2g+(2-2g(X))d-\sum_{i=1}^s l^*(\mu^{(i)})-2)/l^*(\nu)}, \label{cHv1} \end{align}\qquad{(2)}\] where \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_\nu^{X}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers with \(b^X_\nu(\mu^{(1)},\dots,\mu^{(s)},\frac{d!}{z_\nu})=1\).
The following corollary easily follows from Theorem 2.
Corollary 1. For any fixed \(d\geq5,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), the asymptotic behavior of \(H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},\nu,\nu,\dots)\) is given by \[\begin{align} H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},&\nu,\nu,\dots) \sim\frac{2 d!^{s+2g(X)-2}}{z_{\mu^{(1)}} \cdots z_{\mu^{(s)}}}\nonumber\\ &\times\big(\frac{d!}{z_\nu}\big)^{(2g+(2-2g(X))d-2-\sum_{i=1}^s l^*(\mu^{(i)}))/l^*(\nu)},\,\,g\rightarrow \infty. \end{align}\]
This paper is organized as follows: In Sec. 2, we prove Theorem 1 and Theorem 2. Further remarks are given in Sec. 3.
2 Proof of Theorem 1 and Theorem 2↩︎
Our proofs rely on a combinatorial interpretation of the central character \(f_{(r,\,1^{d-r})}(\lambda)\) due to Frumkin-James-Roichman. Recall that for partitions \(\lambda,\nu\vdash d\), the central character \(f_{\mu}(\lambda)\) is defined by \[\begin{align} f_{\mu}(\lambda):=\frac{d!}{z_{\mu}}\frac{\chi_\lambda(\mu)}{\text{dim}\lambda}.\label{f} \end{align}\tag{1}\] Theorem A (Frumkin-James-Roichman [9]). For \(r\leq d\), \(\lambda\vdash d\), \[\begin{align} f_{(r,\,1^{d-r})}(\lambda)=\sum_{\substack{\Gamma\in\{\text{Young trees of}\\\text{order r in diagram \lambda}\}}} (-1)^{\text{vert}(\Gamma)}\cdot \text{weight}(\Gamma). \label{fr} \end{align}\qquad{(3)}\] A Young tree \(\Gamma\) of order \(r\) is a connected, acyclic graph on \(r\) vertices corresponding to distinct boxes of diagram \(\lambda\), in which an edge exists between two vertices if and only if they are in the same row or column with no other vertex of \(\Gamma\) lying strictly between them. A simple path in \(\Gamma\) is a connected subgraph contained entirely within one row or one column. Such a path is called maximal if it is not strictly contained in any other simple path of \(\Gamma\) as a subgraph. The term \(\text{vert}(\Gamma)\) counts the vertical edges in \(\Gamma\), and \[\text{weight}(\Gamma):=\prod_{p } l(p)!\] is the product taken over all maximal simple paths \(p\) in \(\Gamma\), and \(l(p)\) being the number of edges in \(p\). Using this result, we derive the following lemma.
Based on the definition (cf. [3], [4]) and the Lemma 2, we determine the structure of Hurwitz numbers.
Lemma 1. \[\begin{align} \frac{|\chi_\lambda(r,\,1^{d-r})|}{\chi_\lambda(1^d)}\leq \frac{1}{r-1}+\frac{r-2}{(r-1)}\Big(\sum_i^{l(\lambda)}\frac{\binom{\lambda_i}{r}}{\binom{d}{r}}+\sum_i^{l(\lambda')}\frac{\binom{\lambda'_i}{r}}{\binom{d}{r}}\Big).\label{7} \end{align}\qquad{(4)}\]
For any Young tree \(\Gamma\) lying in diagram \(\lambda\), it is obvious that \[\begin{align} \text{weight}(\Gamma)\leq (r-2)! \end{align}\] unless \(\Gamma\) lies in a single row or column; in this case we call \(\Gamma\) a straight Young tree. As there are at most \(\binom{d}{r}\) Young trees in diagram \(\lambda\), \[\begin{align} &|f_{(r,\,1^{d-r})}(\lambda)|\leq\sum_{\substack{\Gamma \text{ is a straight}\\ \text{ Young tree}}} \text{weight}(\Gamma)+\sum_{\substack{\Gamma \text{ is not a straight}\\ \text{ Young tree}}}\text{weight}(\Gamma)\nonumber\\ &\leq \Big(\sum_i^{l(\lambda)}\binom{\lambda_i}{r}+\sum_i^{l(\lambda')}\binom{\lambda'_i}{r}\Big)\cdot (r-1)!+\Big(\binom{d}{r}-\sum_i^{l(\lambda)}\binom{\lambda_i}{r}-\sum_i^{l(\lambda')}\binom{\lambda'_i}{r}\Big)\cdot(r-2)!.\label{4} \end{align}\tag{2}\] By 1 , the result follows.
Lemma 2. If \(d\geq7, 2\leq r\leq d\), partitions \(\lambda\neq(d),(1^d),\vdash d\), then \[\begin{align} \frac{|\chi_\lambda(r,\,1^{d-r})|}{\chi_\lambda(1^d)}\leq&\frac{|d-r-1|}{d-1} ,\qquad r\neq d-1,\label{r}\\ \frac{|\chi_\lambda(d-1,1)|}{\chi_\lambda(1^d)}\leq&\frac{2}{d(d-3)},\label{dMinus1} \end{align}\] {#eq: sublabel=eq:r,eq:dMinus1} where equality in ?? occurs iff \(\lambda=(d-1,1)\) or \((2,\,1^{d-2})\) and equality in ?? occurs iff \(\lambda=(d-2,2)\) or \((2,2,1^{d-4})\).
By symmetry, without loss of generality, we assume \(\lambda_1\geq\lambda'_1\).
a. Case \(\lambda_1\geq\lambda'_1\geq4\) and \(5\leq r\leq d-3\).
We proceed evaluate the number of straight Young tree in diagram \(\lambda\). For the rows \(i\geq2\), we move all the boxes of row \(i\) that lie in columns \(j\geq2\), to the right end of the first row, proceeding row by row. For any \(\lambda\vdash d\), we definite \[\begin{align} \phi_{\lambda} : \text{ \text{boxes in} diagram }\lambda&\longmapsto\text{ \text{boxes in} diagram }(d+1-l(\lambda),\,1^{l(\lambda)-1})\nonumber\\ (i,j) &\longmapsto \left\{ \begin{aligned} &(i,j)&i\text{ or }j=1\\ &(1,\sum\nolimits_{k<i}\lambda_k-i+j) &\text{otherwise } \end{aligned} \right., \end{align}\] where \((i,j)\) represents the box at row \(i\), column \(j\).
Lemma 3. The number of straight Young tree of \(\lambda\) does not decrease under map \(\phi_{\lambda}\).
We consider the three possible types of a straight Young tree \(\Gamma\subset \lambda\).
i. \(\Gamma\) is contained in a single column.
If \(\Gamma\) lies in the first column, then \(\phi(\Gamma)=\Gamma\). If \(\Gamma\) is in a column \(j\) with \(j\geq2\), then \(\phi(\Gamma)\) contained in the first row and forms a straight Young tree in \((d+1-l(\lambda),\,1^{l(\lambda)-1})\).
Figure 1: 
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Figure 2: 
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ii. \(\Gamma\) is contained in row \(i\). And if \(i\geq2\), it has no box at position \((i,1)\).
If \(\Gamma\) lies in the first row, then \(\phi(\Gamma)=\Gamma\). If \(\Gamma\) is in a row \(i\) with \(i\geq2\) and has no box at position \((i,1)\), then \(\phi(\Gamma)\) is contained in the first row and forms a straight Young tree in \((d+1-l(\lambda),\,1^{l(\lambda)-1})\).
Figure 3: 
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Figure 4: 
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iii. \(\Gamma\) is contained in row \(i\) with \(i\geq2\) and has a box at position \((i,1)\).
The \(\phi(\Gamma)\) is not a straight Young tree. However, consider the set \(\phi(\Gamma\setminus (i,1))\cup (1,1)\) which lies in the first row. After restoring the necessary edge
according the definition of a Young tree, it becomes a straight Young tree in \((d+1-l(\lambda),\,1^{l(\lambda)-1})\).
Figure 5: 
.
In each case, ever \(\Gamma\) gives rise to a straight Young tree in \((d+1-l(\lambda),\,1^{l(\lambda)-1})\). Thus, the total number cannot decrease.
Since \(\lambda_1\geq\lambda'_1\geq4\) and \(5\leq r\leq d-3\), the number of straight Young trees is at most \(\binom{d-3}{r}\), in the resulting diagram \((d+1-l(\lambda),\,1^{l(\lambda)-1})\). By Lemma 1 \[\begin{align} \frac{|\chi_\lambda(r,\,1^{d-r})|}{\chi_\lambda(1^d)}\leq\frac{1}{r-1}+\frac{(r-2)(d-r)(d-r-1)(d-r-2)}{(r-1)d(d-1)(d-2)}.\label{1} \end{align}\tag{3}\] Notice that \[\begin{align} &\frac{d-r-1}{d-1}-\Big(\frac{1}{r-1}+\frac{(r-2)(d-r)(d-r-1)(d-r-2)}{(r-1)d(d-1)(d-2)}\Big)\nonumber\\ =&\frac{r}{(r-1)d(d-1)(d-2)}((2 r-5) d^2 +(-3 r^2+2 r+10)d+r^3+r^2-4 r-4).\label{2} \end{align}\tag{4}\] For the knowledge of quadratic functions and \(5\leq r\leq d-3\), we have \[\begin{align} (2 r-5) d^2 +(-3 r^2+2 r+10)d+r^3+r^2-4 r-4>0.\label{3} \end{align}\tag{5}\]
b. Case \(\lambda_1\leq d-3,\,\lambda'_1\leq4\) and \(5\leq r\leq d-3\). The number of straight Young trees is \[\begin{align} \binom{\lambda_1}{r}+\binom{\lambda_2}{r}+\binom{\lambda_3}{r}+\binom{\lambda_4}{r}\leq \binom{d-3}{r}. \end{align}\] By 3 , 4 and 5 , the result follows.
c. Case \(\lambda_1\geq d-2\) and \(5\leq r\leq d-3\). By Theorem A and 1 , the result follows.
d. Case \(r=d-2\). The only diagrams that admit a Young tree are: (1). the diagrams formed by the first row and the first column only; (2) the diagrams that additionally include the boxes at the positions \((2,2)\) or \((2,2),\,(2,3)\) or \((2,2),\,(3,2)\). For \(d\geq7\), by Theorem A and 1 , the result follows.
e. Case \(r=d-1\). The only diagrams that admit a Young tree are: (1). the diagrams formed by the first row and the first column only; and (2) the diagrams that additionally include the box at the position \((2,2)\). By Theorem A and 1 , the result follows.
f. Case \(r=d\). The only diagrams that admit a Young tree are the diagrams formed by the first row and the first column only. By Theorem A and 1 , the result follows.
g. Case \(r=2,\,3,\,4\). The values of \(f_{(r,\,1^{d-r})}(\lambda)\) have explicit formulas [13]: \[\begin{align} f_{(2,\,1^{d-2})}(\lambda)=&\frac{1}{2}\sum_{i=1}^{s(\lambda)} \big(b_i(b_i+1)-a_i(a_i+1)\big);\nonumber\\ f_{(3,\,1^{d-3})}(\lambda)=&\frac{1}{6}\sum_{i=1}^{s(\lambda)} \big(b_i(b_i+1)(2 b_i+1)+a_i(a_i+1)(2a_i+1)\big)-\frac{d(d-1)}{2};\nonumber\\ f_{(4,\,1^{d-4})}(\lambda)=&\frac{1}{4}\sum_{i=1}^{s(\lambda)} \big(b_i^2(b_i+1)^2-a_i^2(a_i+1)^2-(4d-6)(b_i(b_i+1)-a_i(a_i+1))\big).\nonumber \end{align}\] where \(a_j=\lambda'_j-j,\,b_i=\lambda_i-i\) and \(s(\lambda)\) is the number of diagonal boxes in the position \((i,i)\). By 1 , the result follows.
The character formula for Hurwitz numbers of genus \(g(X)\) targets X [13], [14] is \[\label{Hurwitz1} H_d^{X*}(\theta^{(1)}, \dots , \theta^{(n)}) = \sum_{ \lambda\vdash d}(\frac{\dim \lambda}{d!})^{2-2g(X)} \prod_{i=1}^n f_{\theta^{(i)}}(\lambda),\tag{6}\] where \(\dim \lambda\) is the dimension of the irreducible representations of the symmetric group \(S(d)\) corresponding to \(\lambda\).
Proposition 3. For any fixed \(d\geq7,2\leq r\leq d-2,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), when \(k(r-1)+\sum_{i=1}^s l^*(\mu^{(i)})=\text{even}\), \[\begin{align} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)}&,r\,1^{d-r},r\,1^{d-r},\dots)\nonumber\\ =&\frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \sum_{1\leq m\leq \frac{d!}{r(d-r)!}}b^{X*}_r(\mu^{(1)},\dots,\mu^{(s)},m) m^{k}, \label{dHv2} \end{align}\qquad{(5)}\] where \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_r^{X*}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers with
1. \(b_r^{X*}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d!}{r(d-r)!})=1\);
2. \(b_r^{X*}(\mu^{(1)},\,\cdots,\,\mu^{(s)},m)=0\), for \(\frac{(d-r-1)d!}{r(d-1)(d-r)!}<m<\frac{d!}{r(d-r)!}\);
3. \(b_r^{X*}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{(d-r-1)d!}{r(d-1)(d-r)!})=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\).
According to 6 , \[\begin{align} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)}&,\underbrace{r\,1^{d-r},r\,1^{d-r},\dots}_{k})\nonumber\\ &=\sum_{\lambda\vdash d}\big(\frac{d!}{\dim\lambda}\big)^{s+g(X)-2}(f_{(r\,1^{d-r})}(\lambda))^k\prod_{i=1}^s \frac{\chi_\lambda(\mu^{(i)})}{z_{\mu^{(i)}}} .\label{5} \end{align}\tag{7}\] Since Lemma 2, 1 and \(\dim\lambda=\chi_\lambda(1^d)\), we have the structure of disconnected Hurwitz numbers as follows: \[\begin{align} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{r\,1^{d-r},r\,1^{d-r},\dots}_{k})= \frac{2}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} &\sum_{1\leq m\leq \frac{d!}{r(d-r)!}}b_r^{X*}(\mu^{(1)},\dots,\mu^{(s)},m) m^{k} ,\nonumber \end{align}\] where \[\begin{align} b^{X*}_r(\mu^{(1)},\dots,\mu^{(s)},m)=\frac{1}{2}\sum_{\substack{\lambda\vdash d\\|f_{(r\,1^{d-r})}(\lambda)|=m}}(\dim(\lambda))^{2-2g(X)}\big(\operatorname{sgn}(f_{(r\,1^{d-r})}(\lambda))\big)^k\prod_{i=1}^{s}\frac{\chi_\lambda(\mu^{(i)})}{\dim(\lambda)}. \label{top1} \end{align}\tag{8}\] By 1 , we have \[\begin{align} d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_r^{X*}(\mu^{(1)},\dots,\mu^{(s)},m)&=\frac{1}{2}\sum_{\substack{\lambda\vdash d\\|f_{(r\,1^{d-r})}(\lambda)|=m}}(\dim(\lambda))^{2}\Big(\frac{d!}{\dim(\lambda)}\Big)^{2g(X)}\nonumber\\ &\times\big(\operatorname{sgn}(f_{(r\,1^{d-r})}(\lambda))\big)^k\prod_{i=1}^{s}f_{\mu^{(i)}}(\lambda). \end{align}\]
Notice that the central character \(f_\mu(\lambda)\) is an integer [9] and \(\chi_{\lambda}(\mu)=(-1)^{l^*(\mu)}\chi_{\lambda'}(\mu)\) (for details, see [15]). It follows that the quantity \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_r^{X*}(\mu^{(1)},\dots,\mu^{(s)},m)\) is an integer.
When \(k(r-1)+\sum_{i=1}^s l^*(\mu^{(i)})=\text{even}\), by Theorem A, we have \[\begin{align} |f_{(r,\,1^{d-r})}(d)|=&|f_{(r,\,1^{d-r})}(1^d)|=\frac{d!}{r(d-r)!}, \end{align}\] which implies \(b_r^{X*}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d!}{r(d-r)!})=1\).
Similarly, statement 2 and 3 then follow from Lemma 2 and 8 .
By the relationship between connected and disconnected Hurwitz numbers, we have \[\begin{align} \sum_{g,d}&\sum_{\mu^{(1)},\dots,\mu^{(s)}\vdash d}\frac{1}{q!} H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},\nu,\nu,\dots)x^{q} \prod_{i=1}^s p^i_{\mu^{(i)}}\nonumber\\ &\qquad=\log(\sum_{k,d}\sum_{\mu^{(1)},\dots,\mu^{(s)}\vdash d}\frac{1}{k!} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{\nu,\nu,\dots}_{k})x^k \prod_{i=1}^s p^i_{\mu^{(i)}})\label{rlt2} \end{align}\tag{9}\] with \(q=(2g+2d-2-\sum_{i=1}^s l^*(\mu^{(i)}))/l^*(\nu)\). Expanding the right hand side of 9 by Taylor series and taking the coefficients of \(\prod_{i=1}^s p^i_{\mu^{(i)}} x^q\) on both sides, we get \[\begin{align} H^X_{g,d}(\mu^{(1)},&\dots,\mu^{(s)},\nu,\nu,\dots)=H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{\nu,\nu,\dots}_{q})\nonumber\\ &-\frac{1}{2}\sum_{\substack{d_1,d_2\geq1\\d_1+d_2=d\\k_1,k_2\geq0\\k_1+k_2=q}}\sum_{\substack{\omega^{(1)},\dots,\omega^{(s)},\nu^{(1)}\vdash d_1\\\sigma^{(1)},\dots,\sigma^{(s)},\nu^{(2)}\vdash d_2\\\omega^{(i)}\cup\sigma^{(i)}=\mu^{(i)},\text{ for}\\i\in\{1,\dots,s\},\nu^{(1)}\cup\nu^{(2)}=\nu}}\frac{q!}{k_1!k_2!}H^{X*}_{d_1}(\omega^{(1)},\dots,\omega^{(s)},\underbrace{\nu^{(1)},\nu^{(1)},\dots}_{k_1})\nonumber\\ &\qquad\qquad\qquad\qquad\qquad \times H^{X*}_{d_2}(\sigma^{(1)},\dots,\sigma^{(s)},\underbrace{\nu^{(2)},\nu^{(2)},\dots}_{k_2})+\cdots.\label{cHdH} \end{align}\tag{10}\] Taking \(\nu=(r,1^{d-r}),\,2\leq r\leq d-2\), the \(\cdots\) term does not contain \(m^q\) with \(m\geq \frac{(d-r-1)d!}{r(d-1)(d-r)!}\). By Proposition 3, the term \(H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{r\,1^{d-r},r\,1^{d-r},\dots}_{q})\) in 10 has the leading term \[\begin{align} \frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \big(\frac{d!}{r(d-r)!}\big)^{q}, \nonumber \end{align}\] and second leading term \[\begin{align} \frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1) \big(\frac{(d-r-1)d!}{r(d-1)(d-r)!}\big)^{q}. \nonumber \end{align}\] Since Proposition 3 and the binomial theorem and \(d_1,d_2\geq 1,d_1+d_2=d\), the remaining terms in 10 have the leading term \[\begin{align} \frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \prod_{i=1}^{s}m_1(\mu^{(i)}) \big(\frac{(d-1)!}{r\cdot(d-r-1)!}\big)^{q}. \nonumber \end{align}\] and the power of its second leading is less than that of \(H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{r\,1^{d-r},r\,1^{d-r},\dots}_{q})\), which give the statements 1-5 of Theorem 1.
We now turn to the preparation for the proof of Theorem 2.
Theorem B (Flatto-Odlyzko-Wales [8]). For \(d\geq5\), partitions \(\lambda,\mu\vdash d\) with \(\mu\neq(1^d)\vdash d\), \[\begin{align} \frac{|\chi_\lambda(\mu)|}{\chi_\lambda(1^d)}\leq1, \end{align}\] with equality if and only if \(\lambda=(d)\) or \((1^d)\).
Lemma 4. For any fixed \(d\geq7,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), when \(k\cdot l^*(\nu)+\sum_{i=1}^s l^*(\mu^{(i)})=\text{even}\), \[\begin{align} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{\nu,\nu,\dots}_{k})= \frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} &\sum_{1\leq m\leq \frac{d!}{z_{\nu}}}b_\nu^{X*}(\mu^{(1)},\dots,\mu^{(s)},m) m^{k}, \label{dHv1} \end{align}\qquad{(6)}\] where \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_\nu^{X*}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers with \(b_\nu^{X*}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d!}{z_{\nu}})=1\).
According to 6 , \[\begin{align} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{\nu,\nu,\dots}_{k})=\sum_{\lambda\vdash d}\big(\frac{d!}{\dim\lambda}\big)^{s+g(X)-2}(f_{(\nu)}(\lambda))^k\prod_{i=1}^s \frac{\chi_\lambda(\mu^{(i)})}{z_{\mu^{(i)}}} .\label{6} \end{align}\tag{11}\]
Since \(\dim\lambda=\chi_\lambda(1^d)\) and 1 , we have the structure of disconnected Hurwitz numbers as follows: \[\begin{align} H^{X*}_{d}(\mu^{(1)},\dots,\mu^{(s)},\underbrace{\nu,\nu,\dots}_{k})= \frac{2}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} &\sum_{1\leq m\leq \frac{d!}{z_{\nu}}}b_\nu^{X*}(\mu^{(1)},\dots,\mu^{(s)},m) m^{k} ,\nonumber \end{align}\] where \[\begin{align} b^{X*}_\nu(\mu^{(1)},\dots,\mu^{(s)},m)=\frac{1}{2}\sum_{\substack{\lambda\vdash d\\|f_{\nu}(\lambda)|=m}}(\dim(\lambda))^{2-2g(X)}\big(\operatorname{sgn}(f_{\nu}(\lambda))\big)^k\prod_{i=1}^{s}\frac{\chi_\lambda(\mu^{(i)})}{\dim(\lambda)}. \label{top2} \end{align}\tag{12}\]
Notice that \[\begin{align} d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_\nu^{X*}(\mu^{(1)},\dots,\mu^{(s)},m)&=\frac{1}{2}\sum_{\substack{\lambda\vdash d\\|f_{\nu}(\lambda)|=m}}(\dim(\lambda))^{2}\Big(\frac{d!}{\dim(\lambda)}\Big)^{2g(X)}\nonumber\\ &\times\big(\operatorname{sgn}(f_{\nu}(\lambda))\big)^k\prod_{i=1}^{s}f_{\mu^{(i)}}(\lambda). \end{align}\]
Since \(f_\mu(\lambda)\) is an integer and \(\chi_{\lambda}(\mu)=(-1)^{l^*(\mu)}\chi_{\lambda'}(\mu)\), we have \(d!^{2g(X)}\prod_{i=1}^s \frac{d!}{z_{\mu^{(i)}}}b_\nu^{X*}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers.
When \(k\cdot l^*(\nu)+\sum_{i=1}^s l^*(\mu^{(i)})=\text{even}\), since Theorem B and \(\frac{\chi_{(d)}(\mu)}{\chi_{(d)}(1^d)}=1,\frac{\chi_{(1^d)}(\mu)}{\chi_{(1^d)}(1^d)}=(-1)^{l^*(\mu)}\), we have \(b_\nu^{X*}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d!}{z_{\nu}})=1\).
It follows from 10 .
3 Further remarks↩︎
We observe that Lemma 2 holds for \(2\leq r\leq d\), while Theorem 1 only covers the range \(2\leq r\leq d-2\). In this section, we apply Lemma 2 to the case \(r=d-1\) and \(r=d\), thereby obtaining Theorem 4 and Theorem 5, respectively. Moreover, by replacing \(r\,1^{d-r}\) with a general partition \(\nu\), we propose Conjecture 1 as a stronger version of Lemma 2. We then go on to propose Conjecture 2, Conjecture 3 and Conjecture 4. In probability theory, a series of uniform upper bounds on character radios that hold for all irreducible representations \(\lambda\) have been established [16]–[19], which may help to prove Conjecture 1. We note this direction as a possible avenue for further investigation.
Theorem 4. For any fixed \(d\geq7,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), we have \[\begin{align} &H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},(d-1,1),(d-1,1),\dots)\nonumber\\ =&\frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \sum_{1\leq m\leq d(d-2)!}b^X_{d-1}(\mu^{(1)},\dots,\mu^{(s)},m) m^{\frac{1}{d-2}(2g+(2-2g(X))d-2-\sum_{i=1}^s l^*(\mu^{(i)}))}, \label{cHv3} \end{align}\qquad{(7)}\] where \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_{d-1}^{X}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers with
1. \(b^X_{d-1}(\mu^{(1)},\dots,\mu^{(s)},d(d-2)!)=1\);
2. \(b^X_{d-1}(\mu^{(1)},\,\cdots,\,\mu^{(s)},m)=0\) for \((d-2)!<m<d(d-2)!\);
3. \(b^X_{d-1}(\mu^{(1)},\,\cdots,\,\mu^{(s)},(d-2)!)=-d^{2-2g(X)-s}\prod_{i=1}^{s}m_1(\mu^{(i)})\);
4. \(b^X_{d-1}(\mu^{(1)},\dots,\mu^{(s)},m)=0\) for \(2(d-2)(d-4)!<m<(d-2)!\);
5. \(b_{d-1}^{X}(\mu^{(1)},\,\cdots,\,\mu^{(s)},2(d-2)(d-4)!)=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\).
Theorem 5. For any fixed \(d\geq7,s\geq0\) and \(\mu^{(1)},\dots,\mu^{(s)}\vdash d\), we have \[\begin{align} &H^X_{g,d}(\mu^{(1)},\dots,\mu^{(s)},d,d,\dots)\nonumber\\ =&\frac{2d!^{2g(X)}}{d!^2}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}} \sum_{1\leq m\leq (d-1)!}b^X_{d}(\mu^{(1)},\dots,\mu^{(s)},m) m^{\frac{1}{d-1}(2g+(2-2g(X))d-2-\sum_{i=1}^s l^*(\mu^{(i)}))}, \label{cHv4} \end{align}\qquad{(8)}\] where \(d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_{d}^{X}(\mu^{(1)},\dots,\mu^{(s)},m)\) are integers with
1. \(b^X_{d}(\mu^{(1)},\dots,\mu^{(s)},(d-1)!)=1\);
2. \(b^X_{d}(\mu^{(1)},\,\cdots,\,\mu^{(s)},m)=0\) for \((d-2)!<m<(d-1)!\);
3. \(b_{d}^{X}(\mu^{(1)},\,\cdots,\,\mu^{(s)},(d-2)!)=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1)\).
The proof of Theorem 4 and Theorem 5 follow the same lines as that of Theorem 1, and are therefore omitted.
Motivated by [5], based on extensive numerical experiments, we propose the following conjecture.
Conjecture 1. For \(d\geq10\), let \(\lambda,\mu\vdash d\) satisfy \(\lambda\neq(d),(1^d)\). If \(m_1(\mu)\neq1\) and \(\mu\neq(2^{d/2}), (2^{d/2-1},1^2)\), then \[\begin{align} \frac{|\chi_\lambda(\mu)|}{\chi_\lambda(1^d)}\leq&\frac{|m_1(\mu)-1|}{d-1}, \end{align}\] with equality if and only if \(\lambda=(d-1,1)\) or \((2,\,1^{d-2})\). If \(m_1(\mu)=1,\,m_2(\mu)\neq1\) and \(\mu\neq(3^{d/3-1},2,1)\), then \[\begin{align} \frac{|\chi_\lambda(\mu)|}{\chi_\lambda(1^d)}\leq&\frac{2m_2(\mu)}{(d-1)(d-2)}, \end{align}\] with equality if and only if \(\lambda=(d-2,1,1)\) or \((3,\,1^{d-3})\). If \(m_1(\mu)=1,\,m_2(\mu)=0\) and \(\mu\neq(3^{(d-1)/3},1)\), then \[\begin{align} \frac{|\chi_\lambda(\mu)|}{\chi_\lambda(1^d)}\leq&\frac{2}{d(d-3)}, \end{align}\] with equality if and only if \(\lambda=(d-2,2)\) or \((2,2,\,1^{d-4})\).
In Lemma 2 (see Section 2), using the result of Frumkin-James-Roichman [9], we establish Conjecture 1 for the case \(\mu=(r,1^{d-r})\). A natural extension of Conjecture 1 leads us to the following
Conjecture 2. For \(d\geq10,\,m_1(\nu)\geq2\), the coefficients \(b^X_{\nu}(\mu^{(1)},\,\cdots,\,\mu^{(s)},m)\) in Theorem 2 equal to zero for \(\frac{d!m_1(\nu)}{d\cdot z_\nu}<m<\frac{d!}{z_\nu}\) and \(\frac{d!(m_1(\nu)-1)}{(d-1)z_\nu}<m<\frac{d!m_1(\nu)}{d\cdot z_\nu}\), and \[\begin{align} &b^X_{\nu}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d!m_1(\nu)}{d\cdot z_\nu})=-d^{2-2g(X)-s}\prod_{i=1}^{s}m_1(\mu^{(i)}). \end{align}\] Moreover, for \(\nu\neq(2^{d/2-1},1^2)\), \[\begin{align} b_{\nu}^{X}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d!(m_1(\nu)-1)}{(d-1)z_\nu})=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1). \end{align}\]
Conjecture 3. For \(d\geq10,m_1(\nu)=0\) and \(\nu\neq(2^{d/2})\), the coefficients \(b^X_{\nu}(\mu^{(1)},\,\cdots,\,\mu^{(s)})\) in Theorem 2 equal to zero for \(\frac{d(d-2)!}{z_\nu}<m<\frac{d!}{z_\nu}\) and \[\begin{align} b_{\nu}^{X}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{d(d-2)!}{z_\nu})=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1). \end{align}\]
Conjecture 4. For \(d\geq10,m_1(\nu)=1\) and \(\nu\neq(3^{d/3-1},2,1)\) or \((2^{d/2-1},1)\) or \((3^{(d-1)/3},1)\), the coefficients \(b^X_{\nu}(\mu^{(1)},\,\cdots,\,\mu^{(s)})\) in Theorem 2 equal to zero for \(\frac{(d-1)!}{z_\nu}<m<\frac{d!}{z_\nu}\) and \[\begin{align} b_{\nu}^{X}(\mu^{(1)},\,\cdots,\,\mu^{(s)},\frac{(d-1)!}{z_\nu})=(d-1)^{2-2g(X)-s}\prod_{i=1}^{s}(m_1(\mu^{(i)})-1). \end{align}\] Moreover, the coefficients \(b^X_{\nu}(\mu^{(1)},\,\cdots,\,\mu^{(s)})\) in Theorem 2 also equal to zero for \(\frac{2dm_2(\nu)(d-3)!}{z_\nu}<m<\frac{(d-1)!}{z_\nu}\) when \(m_2(\nu)\neq1\), and also equal to zero for \(\frac{2(d-1)!}{z_\nu(d-3)}<m<\frac{(d-1)!}{z_\nu}\) when \(m_2(\nu)=0\).
Acknowledgement↩︎
I would like to thank Di Yang for his advice. This work was supported by NSFC No. 12371254 and CAS NO. YSBR-032.
References↩︎
Email: lxiang1993@ustc.edu.cn.↩︎