Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.


Introduction
Consider the right Cayley graph of the symmetric group S(d), as generated by the full conjugacy class of transpositions. This is a d 2 -regular graded graph with levels L 0 , L 1 , . . . , L d−1 , where L k is the set of permutations with d − k cycles. Each level L k decomposes as the disjoint union of those conjugacy classes in S(d) labelled by Young diagrams with d − k rows.
Let us introduce an edge labelling of the Cayley graph by marking each edge corresponding to the transposition τ = (s t) with t, the larger of the two elements interchanged by τ . This edge labelling was used by Stanley [20] and Biane [1] to study various connections between permutations, parking functions, and noncrossing partitions 1 .
In words, this monotonicity condition states that the labels of the edges traversed in the first k steps of the walk form a weakly increasing sequence.
While elementary to define, the numbers W k,l (α, β) are related to some rather sophisticated mathematics. Let z, t, u, a 1 , a 2 , . . . , b 1 , b 2 , . . . be commuting indeterminates, and form the generating function where p α (A) and p β (B) denote the power-sum symmetric functions in the variables A = {a 1 , a 2 , . . . } and B = {b 1 , b 2 , . . . }, respectively. The series where [X]Y denotes the coefficient of the term X in a series Y . The numbers H 0,l (α, β) were first studied by Okounkov [13], who called them the double Hurwitz numbers. By a classical construction due to Hurwitz -the monodromy construction -H 0,l (α, β) is a weighted count of degree d branched covers of the Riemann sphere by a compact, connected Riemann surface such that the covering map has profile α over 0, β over ∞, and simple ramification over each of the lth roots of unity. The Riemann-Hurwitz formula determines the genus of the covering surface in terms of the ramification data of the covering map: Verifying and extending a conjecture of Pandharipande [18] in Gromov-Witten theory, Okounkov proved that the generating function H(z, 0, u, A, B) is a solution of the 2-Toda hierarchy of Ueno and Takasaki. The 2-Toda hierarchy is a countable collection of partial differential equations, each of which yields a recurrence relation satisfied by the double Hurwitz numbers. A construction of the Toda hierarchy may be found in [15, §4]. Kazarian and Lando [12] subsequently showed that, when combined with the ELSV formula [4], Okounkov's result yields a streamlined proof of the Kontsevich-Witten theorem relating intersection theory in moduli spaces of curves to integrable hierarchies.
Goulden, Jackson and Vakil [10] gave an alternative interpretation of the double Hurwitz number H 0,l (α, β) as counting lattice points in a certain integral polytope. As a consequence of this interpretation and Ehrhart's theorem, it was shown in [10] that, after a simple rescaling, H 0,l (α, β) is a piecewise polynomial function of the parts of α and β, when ℓ(α) and ℓ(β) are held fixed. Detailed structural properties of this piecewise polynomial behaviour were postulated in [10], and subsequently shown to hold by Johnson [11] using the combinatorics of the infinite wedge representation of gl(∞). Shadrin, Spitz and Zvonkine [19] have recently generalized double Hurwitz numbers to the setting of the completed cycle theory introduced by Okounkov and Pandharipande [14]. It is shown in [19] that piecewise polynomiality of double Hurwitz numbers with completed cycle insertions follows from a suitable modification of Johnson's arguments.
The monotone double Hurwitz numbers H k,0 (α, β) were introduced by the present authors in [6], where it was shown that they are the combinatorial objects underlying the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral, an important special function in random matrix theory. Structural properties of the monotone single Hurwitz numbers H k,0 (α) = H k,0 (α, 1 d ) were studied in detail in [7,8], where it was shown that they enjoy a high degree of structural similarity with the classical single Hurwitz numbers H 0,l (α) = H 0,l (α, 1 d ).
In this article, we extend the theorems of Okounkov and Goulden-Jackson-Vakil to the more general setting of the mixed double Hurwitz numbers H k,l (α, β), which interpolate between the classical double Hurwitz numbers (k = 0) and the monotone double Hurwitz numbers (l = 0). Theorem 1. The generating function H is a solution of the 2-Toda hierarchy.
Theorem 2. The mixed double Hurwitz numbers are piecewise polynomial.

Toda equations
Let us group the transposition generators of S(d) into a matrix, Denote by the column sums of this matrix, viewed as elements of the group algebra QS(d).
These elements commute. Set J 1 := 0, and introduce the multiset Let Λ denote the Q-algebra of symmetric functions. We will consider the evaluation of the complete symmetric function h µ ∈ Λ indexed by the (k, l)-hook Young diagram µ = (k, 1 l ) on the alphabet Ξ d . From the definition of the complete symmetric functions, we have The columns sums of T are known as the Jucys-Murphy elements of QS(d). It was observed by Jucys and Murphy (see [3] for a proof) that e r (Ξ d ), the rth elementary symmetric function evaluated on the alphabet of Jucys-Murphy elements, is precisely the sum of the permutations on level L r of the Cayley graph: In fact, since the levels of the Cayley graph generate Z(d), this specialization is surjective [5].
Since C α h (k,1 l ) (Ξ d )C β belongs to the centre of QS(d), we can calculate its character in the regular representation using the Fourier transform. Let (V λ , ρ λ ), λ ⊢ d, be pairwise non-isomorphic irreducible representations of QS(d), so that the map The normalized character of C α h (k,1 l ) (Ξ d )C β in the regular representation of QS(d) is thus where, for any C ∈ Z(d), we denote by ω λ (C) the unique eigenvalue of the scalar operator ρ λ (C) ∈ End V λ , i.e. ρ λ (C) = ω λ (C)I V λ . The eigenvalue ω λ (C) is known as the central character of C in the representation (V λ , ρ λ ). The central character of any conjugacy class C µ is given, in terms of the usual character χ λ µ = Tr ρ λ (σ), σ ∈ C µ , by the formula The central character of any symmetric function f evaluated on Ξ d is simply the evaluation of f on the multiset of contents of the Young diagram λ. This remarkable result is due to Jucys and Murphy, see [3] for a proof.
Recalling that the Schur functions have the expansion where λ ⊢ d, the generating function W may be rewritten as follows: Here Y is the set of all Young diagrams (including the empty diagram) and where for any Young diagram λ and cell in λ, c( ) denotes the content of this cell, i.e. its column index less its row index. By convention, an empty product equals 1.
In order to complete the proof of Theorem 1, we appeal to the following result of Orlov and Scherbin [17], and Carrell [2].  W(z, t, u, A, B) of the mixed double Hurwitz numbers is a diagonal content-product solution, with In particular, Okounkov's generating function H(z, 0, u, A, B) for the classical double Hurwitz numbers is the diagonal content product solution with y k = ze ku , while the generating function H(z, t, 0, A, B) for the monotone double Hurwitz numbers is the diagonal content product solution with

Piecewise polynomiality
To prove Theorem 2, let us begin by formulating precisely the relationship between the numbers W k,l (α, β) and H k,l (α, β).
where, for each d ≥ 1, Then, by the exponential formula, we have for each d ≥ 1, as an identity in Q[[t, u, A, B]]. In this identity, the inner sum is over partitions P 1 ⊔ · · · ⊔ P r of {1, . . . , d} into r disjoint nonempty sets. Equating the coefficient of t k u l l! on either side of this identity, we obtain for each d ≥ 1 and k, l ≥ 0, as an identity in Q[ [A, B]]. Note that the sum on the right hand side depends only on the overall block structure of the set partition P 1 ⊔ · · · ⊔ P r , and not on the internal structure of the individual blocks. We can thus replace the right hand side by a sum over integer partitions, where the coefficient c θ is given by θ i ! and f denotes the surjection ζ j ⊢θj ζ 1 ∪···∪ζ ℓ(θ) =α (η 1 ,...,η ℓ(θ) ) η j ⊢θj The sum on the right hand side receives non-zero contributions from Young diagrams θ whose rows can be obtained by gluing together rows of α, and by gluing together rows of β. This is possible if and only if, for each row θ k of θ, there exist subsets I k , J k ⊆ {1, . . . , d} such that In the case of the one-row Young diagram θ = (d), the required sets are simply where the ellipsis stands for contributions from Young diagrams θ that have at least two rows. These contributions are zero unless the constraint 3.1 is met for each row of θ. We thus conclude that W k,l (α, β) and H k,l (α, β) typically agree, up to a factor of d!.
We can give a geometric interpretation of the above as follows. Fix two positive integers m and n and consider the convex region R m,n = {(x 1 , . . . , x m , y 1 , . . . , y n ) : x 1 ≥ · · · ≥ x m > 0, y 1 ≥ · · · ≥ y n > 0, The hyperplanes W IJ , as I ranges over proper nonempty subsets of {1, . . . , m} and J ranges over proper nonempty subsets of {1, . . . , n}, constitute the resonance arrangement of [11]. A chamber c of the resonance arrangement is a connected component of the complement of a hyperplane W IJ in R m,n . On any chamber c of the resonance arrangement, we have that for all lattice points (α, β) ∈ c.
Theorem 2 claims that for each chamber c there exists a polynomial p k,l c in m + n variables such that H k,l (α, β) = p k,l c (α 1 , . . . , α m , β 1 , . . . , β n ) for all lattice points (α, β) ∈ c. From the above discussion and Section 2, we know that H k,l (α, β) is given by the character formula on any chamber of the resonance arrangement. In order to deduce Theorem 2 from this character formula, we appeal to a recent result of Shadrin, Spitz and Zvonkine [19] which asserts the piecewise polynomiality of a general class of sums of the above form.
Let C Y denote the algebra of all functions Y → C. Following Olshanski [16, Proposition 2.4], we define the algebra A of regular functions on Young diagrams to be the subalgebra of C Y generated by the functions λ → f (Cont λ ), f ∈ Λ, together with the function λ → |λ|. Consider the transform S : C Y → C Y×Y from functions on Young diagrams to functions on pairs of Young diagrams defined by where d = |α| = |β|. This definition assumes that α, β have the same size; if |α| > |β| or vice versa, complete the smaller diagram by adding unicellular rows. The result of Shadrin, Spitz and Zvonkine that we need may be stated as follows: if f ∈ A is a regular function, then S f is piecewise polynomial with respect to the resonance arrangement.
Theorem 4. Given a regular function f ∈ A, positive integers m and n, and a chamber c of the resonance arrangement in R m,n , there exists a polynomial p f c in m + n variables such that S f (α, β) = p f c (α 1 , . . . , α m , β 1 , . . . , β n ) for all lattice points (α, β) ∈ c.
From the character formula 3.2, we see that Theorem 2 follows by applying Theorem 4 with f the regular function λ → h (k,1 l ) (Cont λ ).