Polynomiality of monotone Hurwitz numbers in higher genera
Abstract.
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the JucysMurphy elements, and have arisen in recent work on the the asymptotic expansion of the HarishChandraItzyksonZuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus with ramification specified by a given partition is a polynomial indexed by in the parts of the partition.
Key words and phrases:
Hurwitz numbers, matrix models, enumerative geometry1991 Mathematics Subject Classification:
Primary 05A15, 14E20; Secondary 15B52Contents
1. Introduction
1.1. Classical Hurwitz numbers
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data. The most general case which is commonly studied is that of double Hurwitz numbers , where two points on the sphere are allowed to have nonsimple ramification. That is, for two partitions , the Hurwitz number counts degree branched covers of the Riemann sphere by Riemann surfaces of genus with ramification type over 0 (say), ramification type over (say), and simple ramification over other arbitrary but fixed points (where by the RiemannHurwitz formula), up to isomorphism. The original case of single Hurwitz numbers is obtained by taking , corresponding to having no ramification over .
If we label the preimages of some unbranched point by , then Hurwitz’s monodromy construction [11] identifies bijectively with the number of tuples of permutations in the symmetric group such that

has cycle type , has cycle type , and the are transpositions;

the product is the identity permutation;

the subgroup is transitive; and

the number of transpositions is .
The double Hurwitz numbers were first studied by Okounkov [16], who addressed a conjecture of Pandharipande [17] in GromovWitten theory by proving that a certain generating function for these numbers is a solution of the Toda lattice hierarchy from the theory of integrable systems. Okounkov’s result implies that a related generating function for the single Hurwitz numbers is a solution of the KP hierarchy, and as shown by Kazarian and Lando [12, 13] via the ELSV formula [1], this further implies the celebrated WittenKontsevich theorem [14, 22] relating intersection theory on moduli spaces to integrable systems. These developments, which revealed rich connections between algebraic geometry and mathematical physics, have led to renewed interest in the Hurwitz enumeration problem.
1.2. Monotone Hurwitz numbers
Recently, a new combinatorial twist on Hurwitz numbers emerged in random matrix theory. Fix a pair of normal matrices, and consider the socalled HarishChandraItzyksonZuber integral
where the integration is over the group of unitary matrices equipped with its Haar probability measure. Since is compact, the integral converges to define an entire function of the complex variable . This function is one of the basic special functions of random matrix theory. A problem of perennial interest, whose solution would have diverse applications, is to determine the asymptotics of when are given sequences of normal matrices which grow in a suitably regular fashion.
A new approach to the asymptotic analysis of the HCIZ integral was initiated in [2]. Fix a simplyconnected domain containing on which is nonvanishing. Then the equation
has a unique holomorphic solution on subject to . In [2], we proved that, for , the th derivative of at is given by the absolutely convergent series
with coefficients
where are powersum symmetric functions specialized at the eigenvalues of and is the number of tuples of permutations from the symmetric group such that

has cycle type , has cycle type , and the are transpositions;

the product is the identity permutation;

the subgroup is transitive;

the number of transpositions is ; and

writing each as with , we have .
Clearly, if condition (5) is suppressed, the numbers become the classical double Hurwitz numbers , so can be seen as counting a restricted subset of the branched covers counted by . These desymmetrized Hurwitz numbers were dubbed monotone double Hurwitz numbers in [2].
In this paper, we study the monotone single Hurwitz numbers and prove a monotone analogue of ELSV polynomiality in genus . This result was used in [2] to prove the convergence of under appropriate hypotheses. We also obtain an exact formula for . Before stating these results, we recall our previous work on monotone Hurwitz numbers in genus zero.
1.3. Previous results for genus zero
We introduce the notational convention , where it is understood that for a given partition , the parameters and determine one another via the RiemannHurwitz formula .
In our previous paper [3] on monotone Hurwitz numbers in genus zero, we considered the generating function for monotone single Hurwitz numbers
(1.1) 
as a formal power series in the indeterminates and , where denotes the product . We proved the following result, which gives a partial differential equation with initial condition that uniquely determines the generating function . Our proof is a combinatorial joincut analysis, and we refer to the partial differential equation in this result as the monotone joincut equation.
Theorem 1.1 ([3]).
The generating function is the unique formal power series solution of the partial differential equation
with the initial condition .
The differential equation of Theorem 1.1 is the monotone analogue of the classical joincut equation which determines the single Hurwitz numbers. To make this precise, consider the generating function for the classical single Hurwitz numbers
(1.2) 
As shown in[4, 7], is the unique formal power series solution of the partial differential equation (called the (classical) joincut equation)
(1.3) 
with the initial condition . Note that the classical joincut equation (1.3) and the monotone joincut equation given in Theorem 1.1 have exactly the same differential forms on the righthand side, but differ on the lefthand side, where the differentiated variable is in one case, and in the other.
In [3], we used the monotone joincut equation to obtain the following explicit formula for the genus zero monotone Hurwitz numbers.
Theorem 1.2 ([3]).
The genus zero monotone single Hurwitz number , is given by
where
denotes a rising product with factors, and by convention
Theorem 1.2 is strikingly similar to the wellknown explicit formula for the genus zero Hurwitz number
(1.4) 
published without proof by Hurwitz [11] in 1891 (see also Strehl [20]) and independently rediscovered and proved a century later by Goulden and Jackson [4].
1.4. Main results
In this paper we consider monotone Hurwitz numbers in all positive genera. For genus one, corresponding to branched covers of the sphere by the torus, we obtain the following exact formula.
Theorem 1.3.
The genus one monotone single Hurwitz number , is given by
where is the th elementary symmetric polynomial of the values .
For arbitrary genus , let
(1.5) 
Our main result, stated below, gives explicit forms for these genusspecific generating functions in all positive genera.
Theorem 1.4.
Let be a countable set of formal power series in the indeterminates , defined implicitly by the relations
(1.6) 
where , are formal power series defined by

The generating function for genus one monotone Hurwitz numbers is given by

For , the generating function for genus monotone Hurwitz numbers is given by
where the are rational constants.

For , the rational constant is given by
where is a Bernoulli number.
Note that our proof of LABEL:*thm:main is not just an existence proof; the computations to determine the coefficients are quite feasible in practice if the coefficients for lower values of are known. For example, for genus , we obtain the expression
(1.7) 
For genus , the corresponding expression for is given in Appendix A.
A key consequence of Theorem 1.4 is that it implies the polynomiality of the monotone single Hurwitz numbers themselves.
Theorem 1.5.
For each pair with , there is a polynomial in variables such that, for all partitions , , with parts,
1.5. Comparison with the classical Hurwitz case
For genus one, the explicit formula for the monotone Hurwitz number given in Theorem 1.3 is strongly reminiscent of the known formula for the Hurwitz number, given by
which was conjectured in [7] and proved by Vakil [21] (see also [6]).
The expressions for given in Theorem 1.4 above should be compared with the analogous explicit forms for the generating series
for the classical Hurwitz numbers. Adapting notation from previous works [6, 5, 8] in order to highlight this analogy, let be a countable set of formal power series in the indeterminates , defined implicitly by the relations
(1.8) 
and let , be formal power series defined by
Then, the genus Hurwitz generating series is [6]
and for we have [8]
(1.9) 
where the are rational constants. For example, when we obtain [5]
(1.10) 
For genus , the corresponding expression for is given in Appendix A.
Theorem 1.5 is the exact analogue of polynomiality for the classical Hurwitz numbers, originally conjectured in [7], which asserts the existence of polynomials such that, for all partitions with parts,
(1.11) 
1.6. A possible geometric interpretation?
The only known proof of Equation 1.11 relies on the ELSV formula [1],
(1.12) 
Here is the (compact) moduli space of stable pointed genus curves, , , are (complex) codimension 1 classes corresponding to the marked points, and is the (complex codimension ) th Chern class of the Hodge bundle. Equation (1.12) should be interpreted as follows: formally invert the denominator as a geometric series; select the terms of codimension ; and “intersect” these terms on .
In contrast to this, our proof of Theorem 1.5 is entirely algebraic and makes no use of geometric methods. A geometric approach to the monotone Hurwitz numbers would be highly desirable. The form of the rational expression given in part (ii) of Theorem 1.4, in particular its high degree of similarity with the corresponding rational expression (1.9) for the generating series of the classical Hurwitz numbers, seems to suggest the possibility of an ELSVtype formula for the polynomials . Further evidence in favour of such a formula is obtained from the values of the rational coefficients that appear in these expressions. First, the Bernoulli numbers have known geometric significance. Second, comparing the expressions (1.7) and (1.10) for genus and the expressions in Appendix A for genus gives strong evidence for the conjecture (now a theorem, see [10]) that
(1.13) 
where and are the rational coefficients that appear in LABEL:*thm:main(ii) and (1.9), respectively. But the ELSV formula implies that the coefficients in the rational form (1.9) are themselves Hodge integral evaluations, and for the top terms these Hodge integrals are free of classes—the Witten case. Equation (1.13), which deals precisely with the case , might be a good starting point for the formulation of an ELSVtype formula for the monotone Hurwitz numbers.
1.7. Organization
The bulk of this paper is dedicated to proving parts (i) and (ii) of Theorem 1.4, which give an explicit expression for the generating function and a rational form for , . Part (iii) of Theorem 1.4, which specifies the lowest order term in the rational form for , , follows directly from a result of Matsumoto and Novak [15]. For this reason, we present this proof first, in Section 2.
The necessary definitions and results from our previous paper [3] dealing with the genus zero case are given in Section 3, together with additional technical machinery and results. In Section 4, we introduce a particular ring of polynomials, and establish the general form of a transformed version of the generating function , . In Section 5, we invert this transform, and thus prove parts (i) and (ii) of Theorem 1.4. In Section 6, we use Lagrange’s Implicit Function Theorem to evaluate the coefficients in , and thus prove Theorem 1.3 and Theorem 1.5. Finally, the generating functions and are given in Appendix A.
1.8. Acknowledgements
It is a pleasure to acknowledge helpful conversations with our colleagues Sean Carrell and David Jackson, Waterloo, and Ravi Vakil, Stanford. J. N. would like to acknowledge email correspondence with Mike Roth, Queen’s. The extensive numerical computations required in this project were performed using Sage [19], and its algebraic combinatorics features developed by the SageCombinat community [18].
2. Bernoulli numbers
The computation of the constant term for in Theorem 1.4 relies on a general formula of Matsumoto and Novak (see [15]) for monotone single Hurwitz numbers for the special case of permutations with a single cycle. We give the proof here, as it does not depend on the machinery needed to prove the rest of Theorem 1.4.
Proof of Label:*thm:main(iii).
To compute the monotone single Hurwitz number for a permutation with a single cycle, we can expand the expression for given in Theorem 1.4 as a power series in , and then further expand this as a power series in , throwing away any terms of degree higher than 1 at each step. For the partition consisting of a single part, this yields the expression
For fixed , this expression is times a polynomial in , and evaluating this polynomial at gives . In contrast, according to Matsumoto and Novak’s formula [15, Equation (48)], we have
Again, for fixed , this expression is times a polynomial in . Evaluating this polynomial at gives
since is the exponential generating function for the Bernoulli numbers. ∎
3. Algebraic methodology and a change of variables
3.1. Algebraic methodology
In our previous paper [3] on monotone Hurwitz numbers in genus zero, we introduced three (families of) operators: the lifting operators , the projection operators , and the splitting operators , which involve the indeterminates and a collection of auxiliary indeterminates . These operators were defined by
In terms of these operators, the genusspecific generating functions defined in Equation 1.5 for are characterized by the following result, which is essentially a reworking of the monotone joincut equation given in Theorem 1.1.
Theorem 3.1 ([3]).

The generating function is the unique formal power series solution in the ring of
with the initial condition .

For , is uniquely determined in terms of , , by

For , the generating function is uniquely determined by the generating function and the fact that .
3.2. A change of variables
In [3], where we determined from LABEL:*thm:pdeunique(i), we found it convenient to change variables from and to and via the relations
(3.1) 
and to define the formal power series , by
Expressing the operators , , in terms of and , we obtained
We were also able to show that
(3.2) 
where , and , and, for each , that
Summing this over gives
(3.3) 
where , and .
In terms of these transformed variables, we were able to solve the monotone joincut equation for genus given in LABEL:*thm:pdeunique(i), to obtain [3, Corollary 4.3]
(3.4) 
In this paper we will be solving the monotone joincut equation for genus given in LABEL:*thm:pdeunique(ii). The following result will allow us to reexpress the lefthand side of this equation in a more tractable form.
Proposition 3.2.
For , we have
where is the linear operator defined by
Proof.
From (3.4) and the expression for given above (and using the fact that has no constant term as a power series in ), we have
LHS  
But it is routine to check that
so we have
LHS  
giving the result. ∎
3.3. Auxiliary power series
We also find it convenient to define auxiliary power series related to the power series , and , in which appear in the statement of Theorem 1.4. These are the power series , , and , in , defined by
so that
(3.5) 
3.4. Computational lemmas
The following computational lemmas are used extensively in the rest of the paper to apply the lifting operator to expressions involving the indeterminates and the series .
Lemma 3.3.
For , we have the identity
Proof.
We compute directly the commutator
and the result follows immediately. ∎
Lemma 3.4.
We have
Proof.
The first equation follows directly from the expression for given in Section 3.2. The other two equations can be obtained by applying 3.3 to the expressions in (3.5) for and . ∎