b -MONOTONE HURWITZ NUMBERS: VIRASORO CONSTRAINTS, BKP HIERARCHY, AND O ( N ) -BGW INTEGRAL

A BSTRACT . We study a b -deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We give an evolution equation for this model and derive from it Virasoro constraints, thereby proving a conjecture of Féray on Jack characters. A combinatorial model of non-oriented monotone Hurwitz maps which generalizes monotone transposition factorizations is provided. In the case b = 1 we obtain an explicit Schur expansion of the model and show that it obeys the BKP integrable hierarchy. This Schur expansion also proves a conjecture of Oliveira– Novaes relating zonal polynomials with irreducible representations of O ( N ) . We also relate the model to an O ( N ) version of the Brézin–Gross–Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.


INTRODUCTION
In recent years, the enumerative properties of branched coverings of the sphere by orientable surfaces, that is, the study of Hurwitz numbers, has received a considerable attention [GJ97, ELSV01, GV03, GJV05, OP06, KL07, GPH17].The subject widely intersects the study of map enumeration, which has been active in combinatorics since the 1960s, and the study of matrix models, whose topological expansions give rise to enumerative series of maps or Hurwitz numbers of various kinds (e.g.[LZ04,Eyn16]).A rich class of models in the area is given by the weighted Hurwitz numbers of [GPH17].They are parametrized by a certain weight function, hereafter denoted G(z).Several classical models of enumerative geometry, such as Hurwitz numbers and dessins d'enfants, can be recovered for some choices of G(z).
One of the most studied models among them are monotone Hurwitz numbers, corresponding to the weight function G(z) = 1 1−z .They have a nice combinatorial interpretation in terms of constrained transposition walks in the symmetric group [GGPN14].They have been proved in [Nov20] to give an explicit combinatorial meaning to the 1/N -expansion the Harish-Chandra/Itzykson-Zuber (HCIZ) and Brézin-Gross-Witten (BGW) integrals over U (N ).Additionally, it was shown that they posses rich structural similarities with classical Hurwitz numbers: they are closely related to hypergeometric tau functions [HO15], they satisfy the topological recursion of Eynard-Orantin [DDM17], and the structural similarity between low-genus formulas is remarkable [GGPN13b].
In this paper, we will be interested in deformations of these models by a one-dimensional parameter called b, or β = 2/(1+b).These deformations have two different origins.
The first one lies in the theory of matrix models, where the β-ensembles [Meh04] interpolate between the several types of standard ensembles.In the 1-matrix model, the β-ensemble interpolates between the Gaussian Unitary Ensemble at β = 2, which consists in random Hermitian matrices, and the Gaussian Orthogonal Ensemble at β = 1, which consists in random real, symmetric matrices.For these two special values of β, these models famously give rise to topological expansions which correspond respectively to the enumeration of maps on orientable and on non-oriented surfaces 1 .For arbitrary values of β, the topological expansion over non-oriented surfaces is still valid but the latter are counted with a certain weight which is a monomial in the parameter 2 b = (2 − β)/β, as shown by La Croix [La 09].Although β-deformations of monotone Hurwitz numbers have not been considered prior to [CD22] and this work, an arbitrary-β 2-matrix model was studied in [BEMPF12], based on a β-deformation of the HCIZ integral proposed in [BE09].As for the BGW model, a β-deformation was proposed in [MMS11] in terms of an expansion on Jack symmetric functions, and it coincides with the function we study here (indeed the paper [MMS11] already derived the Virasoro constraints, via a technique of pure gauge limit; see Remark 1 below).
The second origin lies in algebraic combinatorics.In [GJ96], Goulden and Jackson have introduced a b-deformation of the generating function of bipartite maps, obtained by considering the expansion of this function in Schur functions, and replacing them by Jack symmetric functions of parameter α = 1 + b (see below).This model interpolates between the cases of orientable (b = 0) and non-oriented (b = 1) surfaces, and their b-conjecture, a fascinating open problem in algebraic combinatorics, claims that this function still has an explicit topological expansion for arbitrary values of the parameter b.The β-ensemble can be viewed as a special case of the Goulden-Jackson function, thus the previously mentioned topological expansion of La Croix solves a special case of the b-conjecture, see [La 09].
In the recent paper [CD22], the last two authors have extended this b-deformation to the whole family of tau functions of weighted Hurwitz numbers, i.e. for an arbitrary weight function G(z).The corresponding coefficients are shown to enumerate generalized branched coverings of the sphere by non-oriented surfaces, or equivalently some maps on these surfaces called constellations, with an appropriate combinatorial b-weighting scheme.More precisely, the function of [CD22, Section 6] is given by (1) where the second sum is taken over partitions λ of the integer n, where J (b) λ denotes the Jack symmetric function of parameter (1+b) indexed by the partition λ, expressed as a function of its power-sum variables p = (p 1 , p 2 , . . . ) or q = (q 1 , q 2 , . . .), and where the product is taken over all boxes □ of the partition λ, whose b-content is denoted by c b (□).All these notions will be carefully defined in Section 2.1.Here the weight function G(z) = 1 + g 1 z + . . . is a formal power series whose coefficients can be considered as infinitely many additional free variables of the model.
The main result of [CD22] is that the log-derivative of this function has an expansion with positive integer coefficients, which can be interpreted as counting generalized branched 1 We will use the word "non-oriented" to denote a general surface (a real, compact, 2-dimensional manifold without boundary) which may be orientable or not.
We will follow the habits of the combinatorial literature and favour the parameter b over the equivalent parameter β = 2/(1+b), since it is the one which gives rise to nice positive expansions.coverings of the sphere by non-oriented surfaces with certain weights, whose Euler characteristic is tracked by the power of u (see also [Ben21]).This statement is strongly related to the b-conjecture of Goulden and Jackson.In particular it implies a special case of the b-conjecture and it generalizes La Croix's results, see [CD22,Section 6.4] for a discussion.For b = 0, the function τ G b in (1) is the 2-Toda tau function of (orientable) weighted Hurwitz numbers [OS00,GPH17].This naturally calls for studying these b-deformed weighted Hurwitz numbers in more depth, especially for choices of weight functions G(z) which are of special interest due to their connections with other objects in mathematics.Part of this program has actually been completed long before these notions were introduced, in the case of the β-ensemble for the 1-matrix model already mentioned.It corresponds to a linear G(z) with the specialization q i = δ i,1 .This β-ensemble enjoys a collection of nice properties: explicit formulation as matrix integrals at b = 0 and 1 [Meh04], explicit Virasoro constraints [AvM01], explicit combinatorial expansion [CD22], tau function of the KP hierarchy at b = 0 [KMM + 91] and of the BKP hierarchy 3 of Kac and Van de Leur at b = 1 [VdL01].It is important to stress that these properties are not expected to be shared by weighted Hurwitz numbers in general.On the contrary they reflect the importance of this particular choice of weight function.See Appendix A for some comments on the β-ensemble for the 1-matrix model, and for the closely related model of dessins d'enfants.With those motivations in mind and looking for a model which replicates the nice properties of the β-ensemble, we will consider in this paper the "monotone Hurwitz" case of the function τ G b , that is to say with the choice of weight function G(z) = 1 1−z (or a rescaling of it, which we denote τ Z b , see Definition 2.3 or Remark 4).In the usual language of enumerative geometry, we will only consider single (rather than double) Hurwitz numbers, meaning that the variable q i in (1) will be taken to q = 1 := (δ i,1 ).For b = 0, this coincides with the aforementioned notion of single monotone Hurwitz numbers, therefore our object of studies is a non-oriented, b-deformed, extension of them.
We will show that this model essentially enjoys the same nice properties listed above for the β-ensemble, namely relation to matrix integrals, Virasoro constraints, KP and BKP hierarchies, and combinatorial topological expansion.However, as is already the case for b = 0, the case of monotone Hurwitz numbers is arguably more delicate (see e.g. the discussion in Section 5.3).In particular, the techniques developed in this paper are not the same as those of [MMS96,ZJ02,Ale18] for the BGW model (b = 0), [VdL01] for maps at b = 1, [AvM01] for β-ensembles, and [MMS11] for the "pure gauge" limit of Selberg integrals.Those references indeed all rely on matrix integrals while here we instead use algebraic combinatorics and in fact we are eventually able to derive applications to matrix integrals.
Several notions appearing in our analysis give rise to results of independent interest.We make a connection to characters of the orthogonal group for which we find some seemingly new Pfaffian expression.We further prove two conjectures coming from different contexts 3 In the literature, two different hierarchies bear the name "BKP".In this paper we use the "large/charged" BKP hierarchy of Kac and Van de Leur [KvdL98], whose tau functions have expansions on Schur functions with Pfaffian coefficients and come from the orbit of the group O(2∞ + 1) ⊃ GL(∞) on the Dirac vacuum; not the "small/neutral" one of the Kyoto school [DJKM82] which is related to expansions in Q-Schur functions and to the (smaller) group O(∞) ⊂ GL(∞) (and used for example in [Nim90,Orl03]).As it turns out, our functions are not solution of the small BKP hierarchy (nor in fact of the KP hierarchy).[Fér12,ON21] dealing with zonal and Jack polynomials, and orthogonal characters.We also apply our results to evaluate the O(N ) BGW integral in terms of Pfaffians.
We now present the organization of the paper, highlighting the main results.
• In Section 2 we prove an evolution equation, or quantum spectral curve, which is a d dt equation for our main function τ Z b (Theorem 2.4).We use it to prove a set of Virasoro constraints (Theorem 2.7).The evolution equation is a direct consequence of the Virasoro constraints, but in our case it is important to proceed in the other direction, i.e. to deduce the constraints from this equation thanks to a lemma of independent interest (Lemma 2.6).We then use the Virasoro constraints to prove a conjecture of Féray [Fér12] related to Jack characters (Theorem 2.8).
• In Section 3 we introduce a simple combinatorial model of embedded graphs, whose exponential generating function is precisely τ Z b .It directly extends the classical combinatorial model of non-deformed monotone Hurwitz numbers given by non-decreasing factorizations of a permutation into transpositions [GGPN13a].
• In Section 4 we study the function τ Z b for b = 1.We show that this function, which is defined by its expansion in zonal polynomials in this case, has an explicit expansion in Schur functions with rescaled variables, involving dimensions of irreducible orthogonal representations (Theorem 4.1).This fact is strongly related to a conjecture of Oliveira and Novaes [ON21], which we prove as a byproduct (Theorem 4.4).
• In Section 5 we show that the coefficients in this scaled Schur expansion can be expressed as Pfaffians.We deduce that a rescaling of the function τ Z b=1 is a tau function of the large BKP hierarchy (Theorem 5.11).This statement involves a discussion about truncations of expansions and about the role of the charge parameter N = u −1 which has to be interpreted as a formal parameter rather than an integer (as in classical matrix models).
• In Section 6 we show that an orthogonal version of the BGW integral [BG80,GW80] can be expressed by specializing τ Z b=1 .We use results of the previous sections to give an explicit solution to this integral in the special case where the eigenvalues of the external matrix have even degeneracy, in terms of Pfaffians of modified Bessel functions (Theorem 6.2).This solves the problem of finding a closed, explicit formula for the orthogonal BGW/HCIZ integral in the case when external matrices are diagonal [BE09].Finally in Appendix A we collect some comments and analogies with the two other models of arbitrary-β 1-matrix model and dessins d'enfants (equivalently non-oriented general maps and non-oriented bipartite maps).In particular we use the techniques we have developed in the previous sections to make explicit some results on maps and bipartite maps which can be considered as known, even if they are not all explicitly written in the literature.

EVOLUTION EQUATION AND VIRASORO CONSTRAINTS
2.1.Definition of the main function.We start by reviewing some properties of Jack symmetric functions.This material is standard, see [Sta89,Mac95] for complements and for notions which we do not define here.Everywhere in the paper, b is a formal or complex variable, and p = (p i ) i≥1 denotes an infinite family of variables.We denote by p * i the differential operator . .] denote the polynomial algebra over the field of fractions Q(b).It is isomorphic to the algebra of symmetric functions over Q(b) by identifying the variables p with the power-sum basis, see [Sta99].In particular the Laplace-Beltrami operator D b given by (3) acts on the symmetric function algebra.Another (linear) basis of Sym b is given by the monomial symmetric functions which we denote by m λ .
We can identify an integer partition λ of n (denoted λ ⊢ n or |λ| = n) with its Young diagram, which is the union of boxes (x, y) with x ≤ ℓ(λ) and y ≤ λ x , where ℓ(λ) is the number of parts of λ.If □ = (x, y) is a box of λ we write □ ∈ λ and we denote by Definition-Proposition 2.2.There is a unique family of symmetric functions {J (1+b) λ (p)} such that for each partition λ, , < is the dominance order on integer partitions, and where We call them Jack symmetric functions of parameter (1 + b).
We can endow Sym b with a scalar product by defining it on the basis of power-sum symmetric functions where δ µ,ν is the Kronecker delta, z µ := i≥1 i m i (µ) m i (µ)!, and m i (µ) is the number of parts of µ equal to i.A fundamental fact is that Jack symmetric functions are orthogonal, with the following squared norm: λ , where To prevent confusion, we emphasize that the operator p * i in (2) is not the dual of the multiplication by p i for ⟨•, •⟩ b (it is for ⟨•, •⟩ 0 , although we will not directly use this fact).
In this paper we will use the notation ] to denote respectively polynomials, rational functions, and formal power series; for example Q(b)[p][[u, t]] is the ring of formal power the name comes from the fact that for b = 1 this operator is related to the classical Laplace-Beltrami operator as shown in [Jam68]; it is common in the literature to use the same name for the general b-case; see [Sta89] and the comment there following eq.( 11) 5 This quantity is usually called the α-content, and denoted by c α (□), with α = 1 + b.Because the parameter b is more natural than α in our context, we prefer to use this convention.series in t and u whose coefficients are polynomials in the variables p i with coefficients themselves rational fractions in b over Q.We now define our main function, with a slight abuse of notation with respect to (1) since we will not need the variables q anymore.Definition 2.3 (Main function).We let τ Z b (t; p, u) be the generating function of b-weighted single Hurwitz numbers of [CD22] given by (1), with weight function G(z) = (1 + z) −1 , and with the rescaling t → t • u.Explicitly, it is defined by This series is understood as an element of Q(b, u)[p][[t]], and it also belongs to 2.2.Evolution equation.Theorem 6.1 in [CD22], specialized to our case, shows that the function τ Z b (t; p, u) obeys a certain linear differential equation involving unbounded iterates of certain explicit differential operators -corresponding to the fact that the expansion of G(z) = (1+z) −1 as an element of Q[[z]] is infinite.Here we take advantage of the particularly simple form of this function to obtain a much simpler and fully explicit equation.

Theorem 2.4 (Evolution equation). The function τ Z
b is uniquely determined by the following equation: where The special case b = 0 of the above equation is known as the cut-and-join equation for the monotone Hurwitz numbers first proved in [GGPN13a] and reproved in [DBKPS19] using different methods.Our proof reduced to this case gives another proof.It is based on the theory of Jack polynomials, and has a similar flavour as the proof of the decomposition equation in [CD22], but it is not the same.The next proposition collects properties of Jack polynomials that we will need and can be found (explicitly or implicitly) in the seminal work of Stanley [Sta89].

Proposition 2.5 ([Sta89]
).For any partition λ one has where c λ↗µ ∈ Z[b] is a (explicit) polynomial in b with integer coefficients and λ ↗ µ means that µ is obtained from λ by adding a single box.Moreover, for any partition λ ⊢ n one has We can now prove the evolution equation.
Proof of Theorem 2.4.Locally to this proof, we write λ to make the notation lighter.We define also Notice that by Eq. (11).Therefore We can rewrite the first sum as follows: We used Eq.(11) in the first equality, then we changed the order of summation and applied (15) in the second equality.The last equalities follow from (12) and from the fact that the exponent in t counts the degree of the associated symmetric functions.Now, consider the second summand.We have Here, the first equality follows from the characterization of Jack symmetric functions as the eigenfunctions of the Laplace-Beltrami operator and from (11).The second equality is obtained by changing the order of the summation and applying (15).The last equalities are obtained by a direct computation of the action of the Laplace-Beltrami operator on the power-sum symmetric functions and by applying (13) and ( 14).Collecting (17) and ( 18) and comparing them with (16) we obtain It gives directly (8) by using the formula (3) for D b .□ 2.3.Virasoro constraints.We now prove the Virasoro constraints, which can be viewed as a refinement of the evolution equation (8) (the evolution equation is the sum of the Virasoro constraints weighted by p i ).In classical combinatorial models (such as maps and bipartite maps) and for b ∈ {0, 1}, the evolution equation has a simple combinatorial interpretation as a root-deletion procedure, and this combinatorial proof of the evolution equation in fact directly proves the Virasoro constraints.For arbitrary b, things are more complicated (see also [CD22, Rem.5]) and we must proceed differently.The next lemma, which can be used in other models (see Appendix A) says that under mild assumptions one can in fact go backwards and deduce the Virasoro constraints from their weighted sum.
Lemma 2.6 (Virasoro constraints from their sum).Suppose that R is a ring and let F ∈ R[p][[t, t 1 , . . ., t r ]] be a formal power series such that for each n ≥ 0, [t n ]F is a homogenous polynomial in p of degree n, where deg(p i ) := i. Suppose that there exists an operator A such that • ( td dt + s k A)F = 0 for a positive integer k, with s ∈ {t, t 1 , . . ., t r } and A independent of s; • p * i F s=0 = 0 for each positive integer i; where the operators L i−kδs,t := Then, the function F satisfies the following Virasoro constraints: We directly obtain Theorem 2.7 (Virasoro constraints for b-monotone Hurwitz numbers).The function τ Z b satisfies the following Virasoro constraints, for i ≥ 1: Proof of Theorem 2.7.The fact that the family of operators {L Z i , i ≥ 1} satisfy Virasoro relations is a direct check.The statement is then a consequence of the evolution equation (Theorem 2.4) and Lemma 2.6 applied to The reader is refered to Appendix A to see applications of Lemma 2.6 with other values of s and k.It remains to prove the lemma.
Proof of Lemma 2.6.In order to show that L i F = 0 we will prove by induction on n ≥ 0 by assumption.Fix a positive integer n and assume that for each i ≥ 1 − kδ s,t and for each l < n the coefficient [s l ] L i F = 0. We have that On the other hand, the commutator on the LHS can be evaluated as follows.First, since where B is an operator satisfying B = O s (1).Therefore, we have the following equality We now compare the coefficients of s n on the both side of the equation.Using our induction hypothesis and the fact that which implies that [s n ] L i F = 0.This concludes the induction step.□ Remark 1.The Virasoro constraints of Theorem 2.7 have already appeared in the work of Mironov, Morozov and Shakirov [MMS11] as a special limit, called "pure gauge" limit, of a set of differential equations written by Kaneko [Kan93].Those differential equations have furthermore been shown to be solved by some β-deformation of a Selberg integral.The pure gauge limit of this integral could thus provide an integral representation of our function τ Z b .Note that our proof of the Virasoro constraints, which is combinatorial in nature, deals directly with Jack symmetric functions and does not rely on matrix integrals, thus answering a question from [MMS11].
2.4.b-deformed contents and Jack characters.Let C b (λ) := {c b (□) : □ ∈ λ} denote the multiset of b-deformed contents and recall that (non-normalized) Jack characters θ µ (λ) := [p µ ]J λ are the coefficients of Jack symmetric functions expanded in the power-sum basis (see [DF16] for the explanation of this terminology).It was proved by Féray [Fér12, Proposition 4.1] that {θ µ (•)} µ⊢n is a linear basis (over Q(b)) of the space of functions f : P n → Q(b) on the set P n of partitions of size n.In particular, one can express the complete homogenous functions evaluated in b-deformed contents as a linear combination of Jack characters: We now prove a conjecture of Féray showing that the coefficients a k µ have a recursive structure.
Proof.Going back to the Definition 2.3 of τ Z b , we expand □∈λ It is further rewritten as by expanding Jack symmetric functions in the power-sum basis and using the definition of a k ρ .We now extract coefficients starting with where the last equality follows the orthogonality relation which is a consequence of the orthogonality of the power-sum symmetric functions and Jack symmetric functions with respect to ⟨, ⟩ b .Therefore, the coefficient a k ρ can be obtained via the following coefficient extraction Féray suggested that the quantities a k ρ might be interpreted combinatorially, which is known for b = 0, 1 due to the connection with Jucys-Murphy elements [Nov10, ZJ10, Mat10].In the following section we provide such a combinatorial interpretation.

GRAPHICAL MODEL: MONOTONE HURWITZ MAPS
In this section we describe a combinatorial model of graphs whose partition function is τ Z b (t; p, u) (up to a minor rescaling of variables).Note that [CD22, Section 6] provides such a model for general constellations 6 which can be specialized to our case, but the model we describe here is arguably simpler and more explicit in our case.It also covers the model of [GGPN13a], which agrees with our model at b = 0 via classical encoding of permutations factorizations into transpositions by Hurwitz maps [Mos89, Pou97, OP09].In the following definition, we allow maps on the sphere with one vertex, one face and no edges (conventionnally viewed as a 2-cell embedding of the one-vertex graph on the sphere).
Definition 3.1 (Hurwitz map).A labelled Hurwitz map M with n vertices and r edges is a 2cell embedding of a loopless multigraph on a compact surface, with the following properties: (a) the vertices of the map are labelled from 1 to n, and the neighbourhood of each vertex is equipped with an orientation.Moreover each vertex has a distinguished corner called active, and we let c i be the active corner incident to the vertex i.The corner c i is most easily represented as an arrow pointing to i in the corresponding angular sector.(b) the edges of the map are labelled from 1 to r; We let e i be the edge labelled i and M i be the submap 7 of M induced by edges e 1 , e 2 , . . ., e i . 6 General constellations from [CD22] give a combinatorial model for double b-deformed (weighted) Hurwitz numbers, while here we are working with single Hurwitz numbers.In the classical case b = 0 double monotone Hurwitz numbers give a 1/N -expansion of the HCIZ integral and we asked in [CD22, Section 6.5] if our combinatorial expansion extends this result and gives a 1/N -expansion of the β-deformation of HCIZ integral introduced in [BH03].The answer for this question is affirmative, which follows from the expansion of the β-HCIZ integral in Jack symmetric functions derived in [HB06]. 7given a subset of edges E in a map M , the submap of M induced by E is best pictured by viewing M as a ribbon graph, and taking the ribbon graph with the same vertex set as M but keeping only edges (ribbons) in E, keeping all the topological incidences between these edges and the vertices; it possibly lives on a different surface than the map M itself.Edges of label > i may be incident to the vertices a i or b i but they are not represented on this picture.Active corner are represented by arrows.Right: when we want to attach a new edge of maximal label to the vertex n, there is a unique possible corner of attachment around this vertex.Given any other active corner of the map, there are a priori two choices of attachment of the other end of the edge, corresponding to the two sides of the arrow, each corresponding to a given twist of the edge.When this new edge joins two connected components together, property (c3) asserts that only one of these two choices is valid.The degree of a face is its number of active corners.These face degrees form a partition of n called the degree profile of M .
We stress that the local properties (c) in the above definition hold in the map M i , not M .That is to say, for a given i, edges of label greater than i play no role in this constraint.
Remark 3. When the underlying surface is oriented, the constraint (c) is equivalent to the fact that in the map M = M r , vertices are oriented according to surface orientation and that the edge labels around each vertex are increasing, starting from the active corner.This is easily seen using (c) and induction on i.Such maps are often called Hurwitz maps in the map community, we thus give here an non-oriented generalization of this notion.Oriented Hurwitz maps are in bijection with tuples of transpositions by viewing the edge e i = (a i , b i ) as a transposition in S n .In this correspondence, the sequence of active corners around faces of the map correspond to the cycles of the permutation ϕ = e 1 • • • e r .Hence the cycle type of ϕ matches the degree profile of M , which is why these maps are conveniently used as a topological model for (monotone or not) Hurwitz numbers (see [OP09]).
The canonical decomposition of a monotone Hurwitz map M is the following algorithm: (1) if the vertex of maximum label is isolated, erase it; if not, remove the edge of maximum label (note that it has to be incident to that vertex by monotonicity); (2) iterate until no vertex remains.As in [CD22], every time an edge is deleted by the algorithm, we collect a certain b-weight in {1, b} subject to the constraints of Measures of Non Orientability (MON).We refer the reader to that paper for details about MON.Let us just say that a MON is a way to associate a weight to each edge deletion which depends on the topological relation between the edge and the map it is deleted from.In particular, when deleting an edge e results in a map M , the associated b-weight is equal to (Figure 2 where M(n, ℓ) is the set of monotone labelled Hurwitz maps with n vertices and ℓ edges, and where λ(M ) ⊢ n is the degree profile of M .This combinatorial interpretation remains valid at the level of connected objects.Namely, the function (1 + b) ln τ Z b (t; p, u) has an expansion similar to (22), where the sum in the RHS is restricted to connected maps, and without the weight . This follows from the evolution equation of Theorem 2.4 written for (1 + b) td dt ln τ Z b (t; p, u), by induction on the order of u.This was shown already in [CD22] in larger generality but with a different combinatorial interpretation.The coefficients of this function can naturally be called the b-deformed non-oriented monotone single Hurwitz numbers.
Remark 4. The reader may wonder why we chose to work with the function τ Z b (t; p, u) since the rescaled function τ Z b (t; p, u) has nonnegative coefficients and a more natural combinatorial interpretation.This (debatable) choice may become clearer in the next section where u will play the role of an inverse dimension parameter.
Proof.The proof of [GGPN13a] in the orientable case, which interprets the evolution equation (8) (for b = 0) as the deletion of the edge e r , relies on a centrality property of symmetric functions of Jucys-Murphy elements which is not obvious to extend to the non-orientable (let alone, b-deformed) case.We thus need to use a different approach in which the full vertex . of maximal label is removed, rather than a single edge.To do this, we use [CD22, Eq. ( 61)] which reads, specialized to our case where Θ Y and Y + are the operators that substitute y i to p i and to y i+1 , respectively, and where In the rest of the proof we use the notation τ for the RHS of ( 22) and we will prove (23).This will be enough to conclude since this equation characterizes coefficients of τ , inductively.
We assume that (23) holds up to order t n−1 .Every monotone Hurwitz map with n vertices can be constructed from one of size n − 1 by adding the vertex n and all edges incident to it, with increasing edge labels.We now analyse this construction at the level of generating functions; during this analysis we use the variables y i rather than p i to mark the degree of the face containing the active corner of n, and we do not count the contribution of this corner to the face degree (this is similar to [CD22]).The process of adding the vertex n goes as follows: (i) We create the isolated vertex n.The contribution to the generating function is y 0 (since the active corner n is unique in its face but we do not count it by convention) times 1/(1+ b) (since a new connected component is created) times n[t n−1 ]τ (since we choose a map of size n−1 but need to take into account the factor n! (n−1)!coming from generating functions).(ii) We add a certain number (say k ≥ 0) of edges, in increasing label, to the vertex n.The contribution of this step is analyzed below.(iii) We finally account for the contribution of the active corner of label n in its face degree (operator Y + ) and we restore the p-variable weighting for that face (operator Θ Y ).We now analyse the contribution of each iteration in Step (ii).Assume that the current map has i − 1 edges, so we want to add the edge e i to it.By property (c1), we need to attach this edge to the vertex n just before the arrow materializing the active corner (in local orientation).Now, there are several ways to attach the other end of the edge e i : • we attach to a corner in the same face, by splitting it into two faces.By property (c2), for each active corner in the face there is a unique way to attach e i to it in this way, thus splitting the degree of the face into two parts.The contribution to the generating funtion is thus i,j≥1 y i−1 p j ∂ ∂y i+j−1 .
• we attach to a corner in the same face, twisting the edge in order to maintain the same number of faces.Again by property (c2), for each active corner in the face there is a unique way to attach e i in this way; the face degree is unchanged and a weight b is collected in the operation; so the corresponding operator is b • i≥0 y i i∂ ∂y i .• we attach to a corner in a different face, thus merging that face and the current face together and adding degrees.Each choice of an active corner in another face of the map a priori gives two ways to add such an edge.Indeed, we can connect e i to either side of the arrow corresponding to the active corner chosen, and for each choice there is a unique way to twist the edge e i so that (c2) holds (Figure 1-Right).These two choices have the same impact on face degrees but they correspond to contributions of 1 and b to the b-weight.However, this analysis has to be corrected when the chosen corner belongs to a different connected component than n.In that case, only one of the two choices is valid by (c3), but since the number of connected components decreases by one the factor Therefore the two cases (same component or not) give a similar contribution and overall they are taken into account by the operator (1 + b) i,j≥1 y i+j−1 i∂ 2 ∂p i ∂y j−1 .Overall, we see that the contribution of all cases is precisely given by the operator Λ Y .The total contribution of Step (iii) is therefore given by the operator k≥0 (uΛ Y ) k = 1 1−uΛ Y , and this concludes the proof.□ Remark 5.The notion of Hurwitz maps that we introduced here is of independent interest, even without the notion of monotonicity.The (non-necessarily monotone) Hurwitz maps give a combinatorial model for the b-deformed tau function of classical single Hurwitz numbers, i.e. the function τ G b (t; p, q = (δ i,1 ), u) with G(z) = e z introduced in [CD22].To see this, it is better to work directly with the cut-and-join equation of this model [CD22, Eq (67)] (with q i = δ i,1 and ℏ = u in the notation of this paper).Indeed it can be interpreted as describing the deletion of the edge of largest label in a Hurwitz map, similarly as what we did here.We leave details to the reader.
We conclude this section by noting that the coefficients of the function τ Z b (t; p, u), being a special case of the weighted-Hurwitz numbers of [CD22, Section 6], have an interpretation as counting (with weights) certain generalized branched coverings of the sphere by surfaces (orientable or not).It is natural to suspect that, in the same way as we were able in this section to replace in our special case the general model of constellations of [CD22] by a much simpler model (monotone Hurwitz maps), it is possible to obtain an explicit model of generalized branched coverings counted by b-monotone Hurwitz numbers which would be simpler than what results from directly specializing the definitions of [CD22].However, this would lead us too far from our main subject and we will not adress this question in this paper.The purpose of this section is to show that the function τ Z b=1 (t; p, u) defined in (7) by its expansion in zonal polynomials, has in fact an explicit expansion in Schur functions, provided we scale the variables p by a factor of 2.Moreover, this explicit expansion is directly related to irreducible representations of orthogonal groups.See Theorem 4.1 and Theorem 4.4 which establishes the closely related conjecture of Oliveira and Novaes.4.1.Irreducible representations of orthogonal and special orthogonal groups.The representation theory of orthogonal and special orthogonal groups was developed in the pionering work of Weyl [Wey97], see e.g.[FH91] for an introduction.The highest weight irreducible characters o λ of the orthogonal group O(2n) are indexed by partitions λ with ℓ(λ) ≤ n.For λ n = 0 they coincide with the irreducible characters so λ of the special orthogonal group SO(2n), and when λ n ̸ = 0 the restriction of o λ to SO(2n) splits into a direct sum so (λ 1 ,...,λn) + so (λ 1 ,...,−λn) .The dimension of the irreducible representation of SO(2n) of the highest weight λ is given by the following formula due to Weyl (see [Wey97,FH91]), which is a consequence of the Weyl character formula: where A different formula for the dimension of the irreducible representation of the orthogonal group O(n) of the highest weight λ, valid regardless of the parity of n, was given by El Samra and King [ESK79].They proved that where hook λ is the hook-product of the partition λ, which can be defined in previously introduced Jack-theoretic notation by hook λ := hook b=0 (λ) = hook ′ b=0 (λ).The quantity hook λ has a well known representation theoretic interpretation, namely f λ := |λ|!/ hook λ is the dimension of the irreducible representation of the symmetric group indexed by λ, see e.g.[Sta99,FH91].
From (26), the quantity o λ (1 n ) ∈ Q[n] can be considered as a polynomial in n of degree |λ|, which allows us to extend the definition of this "dimension" to the case where n is a formal variable.Below we will use this convention with n = u −1 , so that o λ (1 u −1 ) ∈ Q[u −1 ] and therefore 1/o λ (1 u −1 ) has a valid power series expansion at u = 0, i.e.
Theorem 4.1 (Explicit expansion of τ Z b=1 in scaled Schur functions).The function τ Z b=1 (t; p, u) defined by its expansion (7), has the following expansion in Schur functions of the variables p/2, The proof of Theorem 4.1 consists in showing that the RHS of (27) satisfies the evolution equation (8).In fact, we will directly check each Virasoro constraint since this is not more difficult.For a partition λ and an integer k such that ℓ(λ) ≤ k, we define the rational function The next lemma tells us how to evaluate this rational function on integers.Here and later we use the notation ρ i := λ i − i, with the convention If n > ℓ(λ), we have Proof.Let n > k ≥ ℓ(λ) and note that ρ i = −i if i > k.We split the product in (24) as follows, and for all 1 ≤ p < n The products which do not involve ρ i and ρ j can be performed in terms of factorials, yielding This gives the following expression for so λ (1 2n ), Notice that we derived this formula assuming k < n.In the case ℓ(λ) = n and plugging k = n in the formula above we obtain 2 • so λ (1 2n ) since for any partition λ of length n and it yields the RHS of (24) multiplied by two.Therefore by (25) we have that for n > k, and n = k = ℓ(λ).Plugging the hook-product formula (e.g.[Sta99]) for all k ≥ ℓ(λ), we obtain (28).
Notice now that substituting k = n − 1 in (28) we obtain It has the same form as ( 29), but the bound on the products is n − 1 instead of n, and there is a missing factor 2 in front.It is easy to see that the bound can be extended to n by compensating by a factor 2. Indeed, since ρ n = −n, one has n−1 i=1 ρ i −ρn 2n+ρ i +ρn = 1, and n i=1 Using Eq. (28), we can (and we do) promote the definition of a λ (n) to nonnegative integer k-tuples λ ∈ Z k ≥0 .This is an antisymmetric function of the k parameters ρ i = λ i − i.In what follows it will also be natural to think of the Schur function s λ as a function of these parameters.For an integer partition λ with ℓ(λ) ≤ k, we will thus write We extend this definition antisymmetrically to k-tuples ρ ∈ Z k .In particular s (ρ 1 ,...,ρ k ) vanishes if two of the ρ i are equal.
The following lemma describes the action of Virasoro constraints on scaled Schur functions.Its proof will be more natural after the Boson-Fermion correspondence is introduced, so we postpone the proof to Section 5.3, Lemma 5.6.Lemma 4.3 (Action of the Virasoro operator on scaled Schur functions).Let r ≥ 1, u −1 = 2n and recall the notation L Z r from (20).For any vector ρ ∈ Z k with ρ i ≥ −i and where ϵ i = (0, . . ., 1, . . .0) is the vector with a unique 1 in i-th position.
We can now prove the theorem.
Proof of Theorem 4.1.We will show that the RHS of (27) satisfies the same Virasoro constraints as τ Z b=1 (t; p, u), i.e. it is annihilated by the Virasoro operator L Z r b=1 for each r ≥ 1. Multiplying by p r and summing over r, this will show that it satisfies the evolution equation ( 8) with b = 1, which is enough to conclude since this equation has a unique solution in Fix r ≥ 1, and make the change of variables u −1 = 2n.We will prove that for each partition λ the coefficients of s λ (p/2) in the Schur-expansion of is equal to zero.
We first claim that it is enough to show that for each positive integer k and for each ρ ∈ Z k with ρ i ≥ −i and ρ k−i = i − k for 0 ≤ i < r we have, in previous notation as an identity between rational functions of the formal variable n.Indeed, fix a partition λ and let k ≥ ℓ(λ) + r.From the previous lemma, the coefficient of the Schur function s λ (p/2) in the quantity where the second sum is taken over all partitions ν such that ν − rϵ i ≡ λ (here we write λ ≡ µ if there exists a permutation σ such that λ j − j = µ σ(j) − σ(j) for all j, and if this is the case we write σ = σ λ,µ ).Now, note that there exists i such that ν − rϵ i ≡ λ if and only if there exists j such that λ + rϵ j ≡ ν, and if this is the case the permutations σ ν−rϵ i ,λ and σ λ+rϵ j ,ν are inverse to each other, thus have the same sign.Therefore, using that the symbol a ν is antisymmetric in the ν i − i, the last quantity can be rewritten as which, admitting (31), is equal to zero.
We will now prove the identity (31), between rational functions of n.Note that for a fixed partition λ the LHS of (31) does not depend on the choice of k assuming that n > k ≥ ℓ(λ)+r and that ρ i := λ i − i (with the convention that ρ i = −i for i > ℓ(λ)).This is a consequence of (28) which implies that a λ+ϵ i r (n) = 0 for i > ℓ(λ) + r (since Fix a positive integer k ≥ ℓ(λ) + r.Using (28) we can rewrite the LHS of (31), for n large enough, as where (n) (m) := m i=1 (n − i + 1) denotes the falling factorial.Note that in (32) we have ρ k−i = i − k for 0 ≤ i < r.We will thus treat (32) as a rational function f We will now show by induction on k ≥ r + 1 that for n large enough, the function f n,r is constant as a function of (ρ 1 , . . ., ρ k−r ), and that moreover it is equal to 1 2 δ r,1 .Let k ≥ r + 1.We will study the behaviour of f (k) n,r at all possible poles (including infinity), which we now enumerate.
±2z +r ±2z h̸ =i,j In order to show that there is no pole it is enough to recall that the function f for any positive integer N , where ρ i := −i for k < i ≤ k + N (assuming n > k + N ).Indeed, this is true for arbitrary (ρ 1 , . . ., ρ k−r ) ∈ Z k−r ≥ , and since f n,r is a rational function, it also holds true for arbitrary (ρ 1 , . . ., ρ k−r ) ∈ C k−r .Taking N ≥ j we can see that in fact there is no pole in ρ i = j −1−k−r.• the same argument also shows that there is no pole in If k = r + 1, (33) is a finite quantity (which doesn't depend on any indeterminate).If n−1,r (ρ), where ρ = (ρ 1 + 1, . . ., ρ i + 1, . . ., ρ k−r + 1) and the hat means that the i-th value is ommitted.By the induction hypothesis, if n is large enough this quantity is independent of the ρ i and it is equal to 1 2 δ r,1 .Since we examined all possible poles, we have just shown that the rational function f is bounded, therefore by Liouville's theorem it is a constant, as a function of (ρ 1 , . . ., ρ k−r ).Moreover, when k > r + 1, this constant can be evaluated from the last bullet point (where we used the induction hypothesis), which shows that it is equal 1 2 δ r,1 for n large enough.
Therefore to conclude the induction it only remains to adress the base case k = r + 1.In this case we already know that f (k) n,r is constant, and we use (33) above to evaluate it (for convenience in the notation we choose to evaluate f . In order to compute the last expression, we consider it as a rational function of 2n, f r : C → C and we will apply Liouville's theorem again.The only possible poles of f r (all of them are at most simple) are in 2n = 1+(−1) ϵ j, where 0 ≤ j ≤ r −1 and ϵ ∈ {0, 1}.But for r > 1, the residue Res 2n=1−j f r (x) is null because the terms of the sum corresponding to the indices i and r−1−j −i cancel out, and the same is true for Res 2n=1+j f r (x).We thus obtain which implies that f n+1,r (hence f n,r ) is equal to 1 2 δ r,1 .This concludes the inductive part of the proof.
We have thus proved (31) for infinitely many values of n, therefore it is true as an identity between rational functions, and the proof is complete.□ 4.3.The conjecture of Oliveira and Novaes.An immediate corollary of Theorem 4.1 is the proof of [ON21, Conjecture 1].Before we state it, let us introduce the notation from [ON21] so that we can translate this conjecture into our framework.Let N be a formal parameter and m ≥ 1 be an integer, and define

[N ]
(2) where we recall for reference that Z λ (1 N ) = □∈λ (N + c 2 (□)) and χ λ (1 m )/m! = 1/ hook λ which is the hook length formula.Let furthermore where χ γ is the character of the irreducible representation of the symmetric group S m , and ω λ is the zonal spherical function of the Gelfand pair (S 2m , H m ) with H m the hyperoctahedral group, indexed respectively by partitions γ and λ of m.Irreducible characters of the symmetric group and zonal spherical function of the Gelfand pair (S 2m , H m ) are related to Schur s λ and zonal Z λ symmetric functions, respectively, by their expansions in the power-sum basis We refer to [Mac95] for background.We have Theorem 4.4 ([ON21, Conjecture 1]).The following identity holds: Proof.Theorem 4.1 is an explicit equality between an expansion on zonal symmetric functions and an expansion on scaled Schur functions.Using the notations from [ON21], the hook length formula and specializing t = 1, it reads (34) Consider the standard Hall scalar product ⟨, ⟩ = ⟨, ⟩ 1 on the space of symmetric functions with respect to the variables p ′ := p/2.By definition (see e.g.[Mac95]) we have that We will now take the scalar product of both sides of (34) with s γ (p ′ ).The RHS is equal to To evaluate the LHS, we expand the summand in the power-sum basis which directly leads to Therefore we have and multiplying both sides by (2m)! 2 m gives the desired equality.□ 4.4.On the symplectic case b = −1/2.To conclude this section, we mention that in the same way as the value b = 1 is related to the orthogonal group, the case b = −1/2 is related to the symplectic group.We will not state symplectic analogues of all results of this section since the two values are related by a form of duality.For example, the following symplectic analogue of Theorem 4.1 is in fact a direct corollary of it: Corollary 4.5 (Explicit expansion of τ Z b=−1/2 in Schur functions).The function τ Z b=−1/2 (t; p, u) defined by its expansion (7), has the following expansion in Schur functions of the variables p, where sp λ (1 2n ) denotes the dimension of the irreducible representation of the heighest weight λ of the symplectic orthogonal group Sp(2n).This identity holds in Proof.This identity is obtained by applying the transformation ω 2 (p r ) := 2(−1) r−1 p r on both sides of (27) and using the following classical identities (see [ESK79,Sta89,Mac95]): Using the same techniques applied to (35) we can obtain an analogous result to Theorem 4.4 involving symplectic characters and symplectic zonal spherical functions related to symplectic zonal polynomials J (−1/2) λ in the same way as zonal spherical functions ω λ are related to zonal J (1) λ .polynomials.Details are left to the reader.

PFAFFIANS AND FORMAL LARGE BKP HIERARCHY
5.1.Pfaffians.We will use the following definition for the Pfaffian.
Definition 5.1.The Pfaffian of a skew-symmetric matrix A of even size n is the quantity Pf(A) defined by: We have the following classical Schur's Pfaffian identity.
Lemma 5.2.Let n be an integer and x 1 , . . ., x n be n real variables.Set x n+1 := 0 by convention, and x ij := for n odd.
Notice that the case n odd is a consequence of the case n even with our conventions, since then 1≤i<j≤n x i +x j .We will also need the following minor summation formula for Pfaffians.
Proposition 5.3 will be important to handle matrix integrals in Section 6.For the rest of Section 5 we will only need an elementary special case, namely the fact that for matrices A and B of the same size, one has The following theorem shows that the inverse dimension 1/o λ (1 2n ) has a Pfaffian structure.We write it in terms of the coefficient a λ (n) previously introduced.
Theorem 5.4 (Dimension of Orthogonal representations and Pfaffians).Let n be an integer, and suppose that ℓ(λ) ≤ n.Then we have where and λ n+1 = 0 by convention.
Proof.There are four cases to analyze, depending on whether ℓ(λ) < n or ℓ(λ) = n and whether n is even or odd.In this proof we denote and • Assume first that ℓ(λ) = n and n is even.Using (28) and applying the Schur-Pfaffian identity (37) with We conlude by using the identity (39) for B = (V ij ) 1≤i,j≤n and A = • Assume that ℓ(λ) < n and n is even, so that x n = 0. Using (29) and applying the Schur-Pfaffian identity (37), we obtain with the convention that a 0i = −a i0 = 1/(2i! 2 ) and a 00 = 0.The second line is obtained by re-applying the identity det(B) Pf(A) = Pf(BAB t ) with . The effect is to multiply the coefficients of the last column and of the last row of 1≤i,j≤n by 2. Since • Assume that ℓ(λ) = n with n odd.Using (28) and applying the Schur-Pfaffian identity (37) for n odd, we obtain , so that the entries on the last column and on the last row are • Assume that ℓ(λ) < n with n odd, so that x n = x n+1 = 0. Notice that all elements a ij for i, j ≥ −1 have been fixed using the three previous cases, except for a 0,−1 = −a −1,0 .
Using (29) and applying the Schur-Pfaffian identity (37) for n odd, we obtain Boson-fermion correspondence and proof of Lemma 4.3.We start by quickly reviewing background material on the half-infinite wedge and the boson-fermion correspondence, [KRR13,AZ13].Denote V = i∈Z Ce i the infinite-dimensional complex vector space spanned by linearly independent vectors e i s.For n ∈ Z, the half-infinite wedge of charge n is the vector space, denoted For i ∈ Z, we denote ψ i the wedging operator, (43) and ψ * i the contracting operator defined by its action on basis (44) (the sum contains a single non-vanishing term) and extended by linearity on ∞ 2 V .For positive charges n, the vacuum can thus be written These operators satisfy (45) We also introduce ϕ 0 which acts as the identity on vectors of even charges and minus the identity on vectors of odd charges.Altogether, the wedging and contracting operators and ϕ 0 provide a representation of a Clifford algebra on the half-infinite wedge.
Explicitly, it associates to a vector |v⟩ a polynomial where Γ + (p) := e i>0 p i i J i .Moreover, the boson-fermion correspondence turns the above representations of the Heisenberg and Virasoro algebras on the half-infinite wedge into representations over C[q, q −1 , p 1 , p 2 , . . .], defined by and given explicitly by Note that one can straightforwardly extend this formalism to formal power series.Namely, the boson-fermion correspondence still holds as an isomorphism between C[[q, q −1 , p 1 , p 2 , . . .]] and the space of formal (infinite) linear combinations of ∞ 2 V .Relations (47) and (48) still hold.
The next lemma says that the Virasoro constraints transform nicely under the bosonfermion correspondence.Define the operators on the half-infinite wedge defined by This is a classical result in the representation theory of the semi-infinite wedge.As it is of crucial importance to our main result, we provide a proof of this lemma.
Proof.The proof below is directly borrowed from unpublished notes by Kac.The operators ψ * k , ψ −k+1 annihilate the vacuum of charge 0 for all k > 0 and are called annihilators, while the operators ψ * k , ψ −k+1 for k ≤ 0 are called creators.Annihilators (respectively creators) anti-commute with one another.
Proof of Lemma 4.3.We first notice that (30) is equivalent to by making the change of variables p/2 → p.It can be further rewritten as with the Virasoro generators L r given by (50) (note that r ≥ 1).Since by (53) Schur symmetric functions at charge n are in correspondence with the states |n, λ⟩, we can rewrite the LHS of (54) in its fermionic version via the boson-fermion correspondence at charge n given by Lemma 5.6: By definition of the operators L r , J r we find, Now (using again ρ i = λ i − i) the quantity ψ j ψ * j+r |n, λ⟩ = ψ j ψ * j+r e ρ 1 +n+1 ∧ e ρ 2 +n+1 ∧ . . . is nonzero only if there is an index i such that ρ i + n + 1 = j + r, and in this case it is equal to e ρ 1 +n+1−rδ i,1 ∧ e ρ 2 +n+1−rδ i,2 ∧ . . .= e ρ (i) 1 +n+1 ∧ e ρ (i) 2 +n+1 ∧ . . .with ρ (i) = ρ − rϵ i .Therefore we have (note that necessarily i ≤ k − r) We conclude by applying the Boson-Fermion correspondence in the other direction (note that both e ρ 1 +n+1 ∧ e ρ 2 +n+1 ∧ . . .and the generalized Schur function s ρ are antisymmetric in the coordinates of ρ, and therefore they are in correspondence even when ρ is not in decreasing order).□ 5.3.The large BKP hierarchy and its "formal N " solutions.The following definition of the BKP hierarchy is borrowed from [KvdL98,VdL01].
Definition 5.7 (b ∞ and BKP).The algebra b ∞ (also called quadratic algebra, or spin algebra) is the Lie algebra of quadratic expressions in the ψ i , ψ * j and ϕ 0 , where ψ i , ψ * j are defined by ( 43), (44) and ϕ 0 acts as the identity on vectors of even charges and minus the identity on vectors of odd charges.In other words If a function τ ∈ C[q, q −1 , p 1 , p 2 , . . .] is the image via the boson-fermion correspondence of a vector v which lies in the orbit of the vacuum |0⟩ under the action of B ∞ , then it is said to be a polynomial BKP tau function.Namely, there exists g ∈ B ∞ such that v = g|0⟩ and (56) By an abuse of terminology we will also say that the sequence (τ n,v ) n≥1 is a polynomial BKP tau function.
We will need to extend the notion of BKP tau functions in two ways.The first one, which is straightfoward, is to allow BKP tau functions which are not necessarily polynomials.They can be obtained by using formula (56) for as long as all matrix elements of g are well-defined finite or formal sums.The other, more subtle, extension consists in allowing tau functions in which the parameter n is formal rather than an integer, which will be crucial for us.Indeed, the BKP hierarchy requires to evaluate the tau function at integer values of n, but our main function is not defined for u = 1/(2n) with integer n, since its coefficients have poles.We thus need a certain regularization, which will be performed by considering truncations of the Schur function expansion.This will in turn allow for the definition of formal BKP solutions, i.e. functions which depend on a formal parameter denoted N , instead of n integer.This requires some care since our main function can be made a tau function only after normalizing it by a certain function of n (defined a priori for integer, non-formal, n).In the context of matrix integrals, some of these subtleties are hidden in the fact that the underlying specialization of the variables p to powersums of finitely many eigenvalues realize in some sense this truncation explicitly, see (71) in the next section.The next definition enables us to work at the formal level and without specializations.
Assume that there exists a sequence (α K ) K≥1 tending to infinity, such that for each integer value K ∈ N the coefficients of the function Trunc(τ, α K )(N ) have no pole at N = i for any integer i ≥ K.In particular, the function Trunc(τ, α K )(K) is well defined.
Moreover, assume that there exists a sequence (β N ) such that the sequence is a tau function of the BKP hierarchy and such that for all N ≥ 0 and every odd positive integer k one has for some rational function R k (N ) ∈ Q(N ) and for all N, k ≥ 0 one has for some rational function S k (N ) ∈ Q(N ).Then we say that β N τ (N ) is a formal-N BKP tau function.
In the last sentence, we slightly abuse notation by writing "β N τ (N )" since β N is a function on integers while τ (N ) is a formal object which is not well-defined on integers.Being completely rigorous, it is the pair ((β n ) n≥1 , τ (N )) which encodes the data of the formal-N tau function, but the notation we use should be clear enough in what follows.We have Proposition 5.9 (BKP hierarchy [KvdL98], stated here for formal N ).Let β N τ (N ) be a formal-N BKP tau function.Then for k ∈ N, k ≥ 1 the following bilinear identity holds in where q = (q 1 , q 2 , . . . ) is a vector of formal indeterminates.Here we use the following notation: • h j is the complete homogeneous symmetric function of degree j, • Ď = (kD k ) k≥1 where D r is the Hirota derivative with respect to p r defined as the following bilinear mapping • U (q) = e r≥1 qrDr .
Remark 6.Here it is crucial to note that for fixed k, the quantities (−1) k −1 60) are elements of Q(N ), because of the hypothesis (58) and (59).Therefore (60) is truly a formal identity satisfied by the (formal, unnormalized, non-truncated) generating function τ (N ).A sufficient condition for (58) and (59) to hold is that is a rational function in N .This will be the case for the main function studied in this paper, but not for the cases mentioned in Appendix A.
Proposition 5.9 provides an infinite list of partial differential equations on the function τ (N ) and its shifts by looking at coefficients of the Taylor expansion of (60) in the variables q.For example, the "first" equation of this hierarchy, often called the BKP-equation is obtained by taking k = 2 and extracting the coefficient of q 3 .It is given by where we use the notation F (N ) = log τ (N ) and f i = ∂f ∂p i .Here we have used that Proof of Proposition 5.9.By definition of a formal-N BKP tau function we know that β(N )Trunc(τ, α N )(N ) is a BKP-tau function.Therefore, the fundamental result of Kac and Van de Leur [KvdL98] asserts that (60) holds true provided we replace τ (N ) with Trunc(τ, α N )(N ).Now, fix two partitions λ and µ, and consider the coefficient of [p λ q µ ] in the equation thus obtained.This coefficient is a certain polynomial identity relating a finite number of coefficients of the functions We recall that the coefficients of the function τ (•) are rational functions in N .In particular, taking N large enough (i.e.N > N 0 (λ, µ, k)) we can see that this polynomial identity is an identity between rational functions in N , which holds true for infinitely many values of N .Therefore this polynomial identity in fact holds by replacing each coefficient involved by the corresponding rational function.This is precisely what (60) is saying.□ Theorem 5.11 (BKP hierarchy for non-oriented monotone Hurwitz numbers).The function Proof.This is a direct consequence of the formalism developped in this section, together with the explicit expansion of Theorem 4.1 and the Pfaffian structure of a λ (N ) given by Theorem 5.4, Note in particular that β N satisfies (58) and ( 59

ORTHOGONAL BGW INTEGRAL AND ITS SOLUTION
6.1.The BGW integral over O(n).The Brézin-Gross-Witten integral (BGW integral for short) is a function of a complex, n × n matrix X and its complex conjugate X † , defined as an integral over the unitary group U (n) where dU denotes the normalized Haar measure over U (n).The variable t can be completely re-absorbed in X and X † , but we will keep it to make the link with our function τ Z b (t; p, u).As emphasized in [MMS96], this integral has several interesting expansions.We will focus on the "character" expansion (in the terminology of [MMS96]), meaning that we consider BGW U (n) (X, t) as an element of Q[X][[t]], where Q[X] denotes the ring of polynomials in the matrix elements of X and its complex conjugate.Then, for X ∈ GL n (C), BGW U (n) (X, t) admits an expansion which coincides with τ Z b (t; p, u) at b = 0, u = 1/n and p i = p i (XX † ) := tr((XX † ) i ) being the power-sums associated with the n eigenvalues of XX † .We will describe below this equality, and how this evaluation of τ Z b (t; p, u) makes sense.
For an n × n matrix Y with eigenvalues y 1 , . . the Schur polynomial evaluated on the variables y 1 , . . ., y n (recall that we conventionally parametrize symmetric functions by the power-sums of the underlying variables).We start Nevertheless, it is natural to expect that an analogous formula to (75) exists in the orthogonal case and involves Pfaffians.Our second main result gives such a formula in the special case when X is arbitrary but the eigenvalues of XX t all have even multiplicities.Equivalently, they are x 1 , x 1 , x 2 , x 2 , . . ., x n , x n (we allow x i = x j for i ̸ = j).Theorem 6.2 (The orthogonal BGW integral as a Pfaffian).Assume that XX t ∈ GL 2n (R) has eigenvalues x 1 , x 1 , x 2 , x 2 , . . ., t k+l k − l 4(k + l) For n odd, Pf(a λ i +n−i,λ j +n−j ) 1≤i,j≤n+1 det(X i,λ j +n−j ) 1≤i,j≤n+1 with (tx i ) j for i = 1, . . ., n and j ≥ 0 0 for i = n + 1 and j ≥ 0, and for j = −1 and i = 1, . . ., n 1 for i = n + 1, j = −1.(tx i ) k a kl (tx j ) l • By expanding the matrix integrals on Jack symmetric functions 9 .
The Virasoro constraints for the matrix integrals can be obtained by standard techniques [AvM01].Furthermore, the Virasoro constraints for τ Proof.Theorem 6.1 in [CD22] provides an evolution equation for each model, which can be written explicitly.It is then a direct check that Lemma 2.6 applies again, respectively with s = t and k = 2, 1, respectively.□ From the existence of matrix models, it is standard that the functions τ (1) and τ    Those equations can be obtained starting from (82) and using the Cauchy identity to expand e N i=1 V (x i ;p) on Schur functions.Then one commutes the sum over partitions with the integrals and use the bialternant formula for the Schur polynomials, yielding Z .
The same calculation can be performed with Z (2) b=0 (t; p, N, M ).Notice that both Z (1) b=0 (t; 0, N ) and Z (2) b=0 (t; 0, N, M ) are Selberg integrals which can be evaluated explicitly [Meh04]., using the Cauchy identity for Jack symmetric functions, then commutes the sum over partitions with the integrals and evaluate the integrals using [Kad97,Kan93].This is essentially the method proposed in [MMS11] for the b-deformed BGW integral.

Figure 1 .
Figure 1.Left: the local constraints around the edge e i in a Hurwitz map.Edges of label > i may be incident to the vertices a i or b i but they are not represented on this picture.Active corner are represented by arrows.Right: when we want to attach a new edge of maximal label to the vertex n, there is a unique possible corner of attachment around this vertex.Given any other active corner of the map, there are a priori two choices of attachment of the other end of the edge, corresponding to the two sides of the arrow, each corresponding to a given twist of the edge.When this new edge joins two connected components together, property (c3) asserts that only one of these two choices is valid.
(c) for each i in [1, r], let a i < b i be the two vertices incident to the edge e i .Then in the map M i , the following is true: (c1) In the local orientation around vertex b i , the active corner c b i immediately follows the edge e i (see Figure 1-Left); (c2) the corner which is opposite to c b i with respect to the edge e i is the active corner c a i of vertex a i (see Figure 1-Left); (c3) if the edge e i is disconnecting in M i , then the orientations of the vertices a i and b i are compatible in M i (i.e. they can be jointly extended to a neighbourhood of e i ).If moreover one has b 1 ≤ b 2 • • • ≤ b r , then the map is called monotone.
): • 1 if e joins two different connected components of M , or if e splits a face of M into two distinct faces; • b if e is a twisted diagonal added inside a face of M ; • 1 or b if e joins two distinct faces inside the same connected component of M .Moreover, the b-weight associated to e is 1 if and only if the b-weight associated to its twisted edge ẽ is b.Finally, the b-weight is always 1 when M ∪ {e} is orientable.Moreover, we also ask the b-weight to depend only on the connected component in which the edge is deleted.Figure 2 summarizes what we need to know about measures of nonorientability and b-weights.The product of all b-weights collected by the canonical decomposition is a monomial of the form b ν(M ) , where ν(M ) is an integer associated with the Hurwitz map M .Note that ν(M ) = 0 if and only if M is orientable.Finally, we let cc(M ) be the number of connected components of M .Proposition 3.2.The series τ Z b (t; p, u) := τ Z b (−t/u; p, −u) is the generating function of monotone labelled Hurwitz maps in the sense that

Figure 2 .
Figure 2. How the b-weight is computed.In the last two cases, the weights of 1, b or b, 1 are decided arbitrarily in general, but if one of the two pictures is orientable, then it must get the weight 1..

.
4. SCHUR EXPANSION FOR b = 1 AND ORTHOGONAL GROUP CHARACTERSIn this and the next sections, we consider the case b = 1 for which Jack polynomials coincide with the zonal polynomials,Z λ (p) := J (b) λ (p) b=1They are zonal spherical functions for the Gelfand pair (GL(N ), O(N )), which explains the terminology (see[Mac95, Chap.7]).

,
Pf(a λ i +n−i,λ j +n−j ) 1≤i,j≤n+1 det (tx i ) λ j +n−In order to apply the minor summation formula for Pfaffians, we have to promote det(xλ j +n−j i) 1≤i,j≤n to an (n+1)×(n+1) determinant, by adding a row and a column of zeroes to (x λ j +n−j i ) 1≤i,j≤n except for a 1 on the diagonal.Therefore BGW O(2n) (X, t)