Multispecies Weighted Hurwitz Numbers

The construction of hypergeometric $2D$ Toda $\tau$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.


Introduction
In [6,7] a simple method was developed for constructing parametric families of 2D Toda τ -functions [23][24][25] of hypergeometric type [20] that serve as generating functions for the weighted enumeration of n-sheeted branched coverings of the Riemann sphere. These are characterized by the fact that their expansions in the basis of products of Schur functions have only diagonal coefficients, and these are of special content product form. When they are expanded instead in the basis of products of power sum symmetric functions, the coefficients turn out to be weighted sums of Hurwitz numbers, with weighting dependent generally on an infinite sequence of parameters c = (c 1 , c 2 , . . . ) determined by an associated weight generating function G(z). Such weighted sums may be interpreted equivalently as weighted enumeration of paths in the Cayley graph of the symmetric group S n generated by transpositions.
The special choice G(z) = exp(z) gives rise to the generating functions for simple and double Hurwitz numbers introduced by Pandharipande [21] and Okounkov [19], in which all branch points other than a single one or a specified pair have simple branching profiles. It has been known since the pioneering works of Hurwitz [3,4], Frobenius [11,12] and Schur [22] that these numbers may be reinterpreted combinatorially using the monodromy representation of the fundamental group of the punctured sphere with values in S n determined by lifting closed paths to the covering surface. From this viewpoint, they enumerate factorizations of the identity element into products of elements whose conjugacy classes correspond to the ramifications profiles, and hence give uniformly weighted enumeration of paths in the Cayley graph from one conjugacy class to another with a given number of steps.
Another choice of weight generating function that was studied in [7] is the quantum dilogarithm function [2]. This amounts to equating the parameters c i to powers of a single quantum deformation parameter q., and gives rise to four special versions of quantum weighted Hurwitz numbers whose distribution functions were linked to those of a Bosonic gas with linear energy spectrum in [7]. Using a suitably extended class of weight generating functions, this was further generalized in [8,9] to include both the infinite family of classical weighting parameters c and a pair (q, t) of quantum deformation parameters characterizing Macdonald polynomials.
In the present work, the notion of weighted Hurwitz numbers is extended to weighted enumerations of multispecies coverings involving arbitrary choices for the corresponding weighting parameters. The generating function depends on l + m expansion parameters (w 1 , . . . , w l ; z 1 , . . . , z m ), corresponding to two classes and l + m subspecies (or "colours") of branch points with ramification profiles types {µ β } β=1,...,m , {µ α } α=1,...,l . The special case of signed multispecies enumeration in the uniformly weighted case was studied in [10]. Its combinatorial significance was explained in terms of enumeration of paths in the Cayley graph that are subdivided into strictly or weakly monotonically increasing subsequences of transpositions having given lengths. In the single species case [7], this was shown equivalent to (signed) enumeration of branched covers of the Riemann sphere with the "coloured" branch points constrained to have fixed total ramification index within each class. Another special case detailed here consists of "multispecies quantum weighted Hurwitz numbers", in which the weighting parameters consist of powers of a sequence of auxiliary quantum deformation parameters q = (q 1 , . . . , q l ), p = (p 1 , . . . , p m ).
In Subsections 1.1 -1.4 the basic notions regarding Hurwitz numbers will be recalled, together with the construction of weighted Hurwitz numbers using infinite parameter families of weight generating functions G(z), as developed in [6,7,10]. In Section 2, the single species case will be extended to multispecies by introducing the idea of "coloured" branch points, of two classes. Weight generating functions depending on a multiparametric set of expansion parameters multiplicatively, provides a multiparametric family of 2D Toda τfunctions of hypergeometric type that are generating functions for multispecies weighted Hurwitz numbers. As in the single species case, these may be viewed both geometrically and combinatorially, in terms of weighted coverings and paths in the Cayley graph. In Section 3 this is restricted to the special cases of quantum weightings as introduced in [7] to produce generating functions for multispecies quantum Hurwitz numbers. These are interpreted, as in the single species case, both geometrically and combinatorially † .

Hurwitz numbers
For a set of k ∈ N + partitons (µ (1) , . . . µ (k) ) of n ∈ N + , let H(µ (1) , . . . µ (k) ) denote the number of n-sheeted branched coverings of the Riemann sphere (not necessarily connected), with k branch points, whose ramification profiles are given by the partitions (µ (1) , . . . µ (k) ), weighted by the inverse of the order of the automorphism group. These are the so-called (geometrically defined) Hurwitz numbers [11,14]. The genus g of the covering surface is determined by the Riemann Hurwitz formula for the Euler characteristic is the colength of the partition µ; i.e., the complement of the length ℓ(µ) with respect to its weight |µ|, or the degree of degeneracy of the branched cover over a point with ramification profile type µ. The Frobenius-Schur formula [3,4,14,22] expresses these as sums over irreducible S n characters.
where χ λ (µ) is the character of the irreducible representation of symmetry type λ, evaluated on the conjugacy class cyc(µ) consisting of elements with cycle lengths equal to the parts of the partition µ, is the product of the hook lengths in the Young diagram of partition λ and is the order of the stabilizer under conjugation of any element of the conjugacy class cyc(µ), with m i (µ) denoting the number of parts of the partition µ equal to i, There is an alternative interpretation of H(µ (1) , . . . µ (k) ) that is purely group theoretical, and combinatorial; it equals the number of ways in which the identity element I ∈ S n may be expressed as a product of k elements belonging to the conjugacy classes of cycle type {cyc(µ (i) )} i=1,...,k I = g 1 , g 2 · · · g k , where g i ∈ cyc(µ (i) ). (1.6) consisting of BKP generating functions of hypergeometric type, was considered in [17,18].
The two are related by noting that each such factorization may be understood as defining the image under the monodromy map from the fundamental group of the punctured sphere with the branch points removed into S n of the homomorphism from the fundamental group of the sphere, punctured at the branch points, obtained by lifting closed loops to the covering surface.

Weighted geometrical Hurwitz numbers
As defined in [7], given a weight generating function G(z) expressible as an infinite product the weight W G (µ (1) , . . . , µ (k) ) assigned to a configuration of k +2 branch points with ramification profiles µ (1) , . . . , µ (k) , µ, ν, is solely determined by the colengths {ℓ * (µ (1) ), . . . , ℓ * (µ (k) )}, and is given by evaluation of the monomial sum symmetric functions at the parameter values Here λ is the partition of length ℓ(λ) = k whose parts are {ℓ * (µ (i) )} i=1,...,k , and | aut(λ)| is the order of the automorphism group of λ | aut(λ)| := were m i (λ) is the number of parts of λ equal to i. The weighted geometrically defined double Hurwitz numbers H d G (µ, ν) for n sheeted branched coverings of the sphere, give a weighted enumeration of n-sheeted branched covers of the Riemann sphere that contain a pair of fixed branch points, say at (0, ∞), with ramification profile types given by the pair of partitions (µ, ν) and a further set of k branch points with ramification profiles (µ (1) , . . . , µ (k) ). They are defined to be the weighted sums over all k-tuples of nontrivial ramification profiles with weight given by W G (µ (1) , . . . , µ (k) ), satisfying the condition The Riemann-Hirwitz formula for the genus g of the covering surface is then An alternative is to use the dual weight generating functioñ for which the geometrical weight WG(µ (1) , . . . , µ (k) )) is given by the "forgotten" symmetric function f λ (c) (1.14) where the partition λ is again defined as above, with parts consisting of the colengths {ℓ * (µ (i) )} i=1,...,k . The dually weighted geometrical Hurwitz numbers are similarly defined by the weighted sum

Weighted combinatorial Hurwitz numbers
Following [6,7], we may alternatively define a combinatorial Hurwitz number F d G (µ, ν) that gives the weighted enumeration of d-step paths in the Cayley graph of the symmetric group S n generated by transpositions (a, b), b > a starting at an element h ∈ cyc µ in the conjugacy class cyc(ν) consisting of elements with cycle lengths equal to the parts of µ and ending in the conjugacy class cyc(ν) Every such path has a signature λ, which is defined to be the partition of weight d, whose parts are, in weakly decreasing order, the number of times any given second element b i , i = 1, . . . , ℓ(λ) is repeated. In the case of the weight generating function G(z), we assign to any path with signature λ a combinatorial weight equal to the product e λ (c) of the elementary symmetric functions [15], evaluated at the parameters (c a , c 2 , . . . ) In the case of the dual generating functionsG(z), a combinatorial weight equal to the product h λ (c) of the complete symmetric functions [15], evaluated at the parameters (c a ,c 2 , . . . ) Let m λ µν be the number of d = |λ| step paths of signature λ starting at h ∈ cyc(µ) and ending in the conjugacy class cyc(ν). Then the combinatorial weighted Hurwitz numbers In [7] it is proved that these two notions of weighted Hurwitz numbers in fact coincide: The main idea behind the proof is to define associated elements G n (w, with eigenvalues pf the following content product form: ( 1.25) On the other hand, the Cauchy-Littlewood generating function relation [15] and its dual show that G n (z, J ) andG n (z, J ) may be expanded in terms of dual bases of the algebra of symmetric functions, evaluated either on the parameters c or on the Jucys -Murphy elements J = (J 1 , . . . , J n ).
where e λ , h λ , m λ and f λ are the elementary, complete, monomial and "forgotten" symmetric functions [15], respectively. Applying (1.27) and (1.27) to the basis for Z(C[S n ]) consisting of the cycle sums and using the identities The bases {F λ } |λ|=n and {C µ } |µ|=n are related by where χ λ (µ) denotes the irreducible character of the irreducible representation of type λ evaluated on the conjugacy class cyc(µ). Under the characteristic map, this is equivalent to the Frobenius character formula [15] (1.32)

Hypergeometric 2D-toda τ -functions as generating functions
As shown in [7], for any given generating function of type G(z) orG(z), there is a naturally associated 2D Toda τ -function of hypergeometric type, expressible as a diagonal double Schur function expansion are the 2D Toda flow variables, which may be identified in this notation with the power sums in two independent sets of variables. (See [15] for notation and further definitions involving symmetric functions.) The coefficients have the standard content product form that characterize such 2D τ -functions of hypergeometric kind: (1.38) and identical formulae for G replaced byG.

2D Toda τ -functions as generating functions for multispecies weighted Hurwitz numbers
For any choice of weight generating functions G 1 (w 1 ), G 2 (w 2 ), . . . ,G 1 (z 1 ),G 2 (z 2 ), . . . , we may form composites by using the product α G α (w α ) βG β (z β ) as generating function for multiple weighting types. The resulting content product coefficients r α G α (wα) βG β (w β λ are just the product α r (2.1) We may also include weight factors in which some or all of the parameters (z 1 , z 2 . . . ), (w 1 , w 2 . . . ) are repeated in the product. This only affects the constraints on the sums of the colengths in the weighted multispecies Hurwitz numbers. (See, e.g. example 3.3 in [7], in which the weights are uniform, but the linear generating function that gives Hurwitz numbers for Belyi curves and strictly monotonic paths is replaced by a power of the latter, resulting in multiple branch points, with the total colength fixed, and multimononic paths.)

Multispecies combinatorial weighted Hurwitz numbers
The combinatorial multispecies Hurwitz number F (d,d) G(l,m) (µ, ν) is determined as follows. Let D n be the number of partitions of n and lett F dα G α , Fd β G β denote the D n × D n matrices whose elements are F dα G α (µ, ν) and Fd β G β (µ, ν), respectively, as defined in (1.19), (1.20).
for each generating function G α (w α ) orG β (z β ). Since the central elements {G α (w α , J ),G β (z β , J )} all commute, it follows that so do the matrices {F dα G α , Fd β G β }. Denoting their product, in any order, l + m parts, each of which is a subsequence assigned a "colour" and a "class" with l of them of the first class and m of the second. The number of partitions of first class with colour α is d α while the number of second class with colour β isd β . The partitions λ (α) of weights d α are defined to have parts {λ α uα } uα=1,...,kα equal to the number of a transposition appears within that subsequence, having the same second element, and similarly for {λ β v β } v β =1,...,k β with k α = ℓ(λ (α) ),k β = ℓ(λ (β) ) (2.25) the number of such parts. If a weight is given that is equal to the product G(l,m) (µ, ν) is the sum of these, each multiplied by the number of elements of the equivalence class of paths with the given multisignature.
The multispecies generalization of (1.21) is equality of the geometric and combinatorial Hurwitz numbers: Proof. Applying the central element G (l,m) n (w, z, J ) defined in (2.9) to the cycle sum C µ and applying (1.26) for each factor in the product gives (2.29) and the similar formula forG. Comparing this with the eigenvalue formula (2.10) and eq. (2.20) proves the result.

Quantum Hurwitz numbers
Amongst the examples of weighted Hurwitz numbers studied in [7], four special classes were introduced in which the generating functions G(z),G(z) were chosen as a variant of the quantum diloogarithm [2]. This meant that the parameters {c i } were chosen as powers of a quantum deformation parameter q. Suitable interpretation of the parameter q in terms of Planck's constant and Boltzmann factors for a Bosonic gas with linear energy spectrum, leads [7] to a relation between the resulting weighted counting of branched cover and the energy distribution for a Bosonic gas, which further justifies terming these "quantum" Hurwitz numbers.
The quantum Hurwitz numbers introduced in [7] have four variants for which 1-parameter families of 2D Toda τ -function generating functions were constructed. In the first case, the weight generating function is: and hence the parameters c i are identified as {c i := q i−1 } i∈N + . The second is a slight modification of this, with weight generating function and hence the zero power q 0 is omitted, and {c i : The third case is based on the weight generating function

5)
and hence is the dual of the first case, with {c i := q i−1 } i∈N + . The final case is a hybrid, formed from the product of the first and third for two distinct quantum deformation parameters q and p, with weight generating function These are all expressible as exponentials of the quantum dilogarithm function H(q, z) = e Li 2 (q,z) , Q(q, p, z) = e Li 2 (p,z)−Li 2 (q,−z) , . (3.11) The content product formulae for the first and third of these are The associated hypergeometric 2D Toda τ -functions have diagonal double Schur function expansions with these as coefficients: Using the Frobenius character formula (1.32), and setting N = 0, these may be rewritten as double expansions in the power sum symmetric functions [7]: The coefficients are the corresponding quantum Hurwitz numbers H d E(q) (µ, ν), H d H(q) (µ, ν), which count weighted n-sheeted branched coverings of the Riemann sphere, defined by (1) , . . . , µ (k) )H(µ (1) , . . . , µ (k) , µ, ν). (3.19) where the weightings for such covers with k additional branch points are [7]: , (3.20) is over all k-tuples of partitions having nontrivial ramification profiles that satisfy the constraint k i=1 ℓ * (µ (i) ) = d, and H(µ (1) , . . . , µ (k) , µ, ν) is the number of branched n-sheeted coverings, up to isomorphism, having k + 2 branch points with ramification profiles (µ (1) , . . . , µ (k) , µ, ν).
The combinatorial interpretation of the quantum Hurwitz numbers F d E(q) (µ, ν) and F d H(q) (µ, ν) appearing in (3.17) is as follows. Let (a 1 b 1 ) · · · (a d b d ) be a product of d transpositions (a i b i ) ∈ S n in the symmetric group S n with a i < b i , i = 1, . . . , d. If h ∈ S n is in the conjugacy class cyc(µ) ⊂ S n , we may view the successive steps in the product as a path in the Cayley graph generated by all transpositions, whose signature is the partition λ of d, |λ| = d, whose parts λ i consist of the number of transpositions (a i b i ) sharing the same second element. If we further require that the ones with equal second elements be grouped together into consecutive subsequences in which these second elements are constant, with the consecutive subsequences strictly increasing in their second elements, then the number N λ of elements with signature λ is related to the number N λ of such ordered sequences bỹ Denote the number of such paths from the conjugacy class of cycle type cyc(µ) to the one of type cyc(ν) having signature λ asm λ µν , and the number of ordered sequences of type λ as m λ µν . Thusm For all paths of signature λ we assign the weights to paths in the Cayley graph, and obtain the pair of corresponding combinatorial weighted Hurwitz numbers  (3.29) that give the weighted enumeration of paths, using the weighting factors E λ (q) and H λ (q) respectively for all paths of signature λ.
As shown in general in [7], the enumerative geometrical and combinatorial definitions of these quantum weighted Hurwitz numbers coincide: A similar result holds for weights generated by the function E ′ (q, z), with corresponding quantum Hurwitz numbers defined by where the weights W E ′ (q) (µ (1) , . . . , µ (k) ) are given by .

(3.33)
Choosing q as a positive real number, parametrizing it as and identifying the energy levels ǫ k as those for a Bose gas with linear spectrum in the integers, as for a 1-D harmonic oscillator we see that if we assign the energy to each branch point with ramification profile of type µ, it contributes a factor n(µ) = 1 e βǫ(µ) − 1 (3.37) to the weighting distributions, the same as that for a bosonic gas.

3.2
The multiparameter family of τ -functions τ Q(q;w;p,z) (N, t, s) We now consider a multiparameter family of weight generating functions Q(q, w; p, z) obtained by taking the product of any number of the generating functions E(q i , z i ) and H(p j , w j ) for distinct sets of generating function parameters w = (w 1 , . . . , w l ), z = (z 1 , . . . , z m ), and quantum parameters q = (q 1 , . . . , q l ), p = (p 1 , . . . , p m ) Following the approach developed in [7], we define an associated element of the center Z(C[S n ]) of the group algebra C[S n ] by 3.4

Multispecies combinatorial quantum Hurwitz numbers
Another basis for Z(C[S n ]) consists of the cycle sums where cyc(µ) denotes the conjugacy class of elements h ∈ cyc(µ) with cycle lengths equal to the parts of µ. The two are related by These consist of a partition of the d steps into l + m parts, each of which is a subsequence assigned a "colour" and a "class" with l of them of the first class and m of the second. The number of partitions of first class with colour α is d α while the number of second class with colour β isd β . The partitions λ (α) of weights d α are defined to have parts {λ (α) uα } uα=1,...,kα equal to the number of a transposition appears within that subsequence, having the same second element, and similarly for {λ (β) v β } v β =1,...,k β with k α = ℓ(λ (α) ),k β = ℓ(λ (β) ) (3.56) the number of such parts. For each such subpartition, the weight assigned is the product of the weights for each subsegment: l α=1 E λ (α) (q α , w α ) m β=1 H( λ) β (p β , z β ) (3.57) and F (c,d) (q,p )(µ, ν) is the sum of these, each multiplied by the number of elements of the equivalence class of paths with the given multisignature. The multispecies generalization of (3.30) is equality of the geometric and combinatorial Hurwitz numbers: Comparing this with eq. (2.20) proves the result.

Remark 3.1. Multispecies Bose gases
Returning to the interpretation of the quantum Hurwitz weighting distributions in terms of Bose gases, if we choose each q i to be a positive real number with q i < 1, and parametrize it, as before, q i = e −β ω i which are those of a multispecies mixture of Bose gases.