2D Toda tau-functions as combinatorial generating

2D Toda τ -functions as combinatorial generating functions∗ Mathieu Guay-Paquet1 and J. Harnad2,3 1

Universit´e du Qu´ebec ` a Montr´eal 201 Av du Pr´esident-Kennedy, Montr´eal QC, Canada H2X 3Y7 email: [email protected]

arXiv:1405.6303v1 [math-ph] 24 May 2014

2

Centre de recherches math´ematiques, Universit´e de Montr´eal C. P. 6128, succ. centre ville, Montr´eal, QC, Canada H3C 3J7 e-mail: [email protected] 3

Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montr´eal, QC Canada H4B 1R6

Abstract Two methods of constructing 2D Toda τ -functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group Sn by multiplication of the conjugacy class sums Cλ ∈ C[Sn ] in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product C[Sn ] ⊗ C[Sn ] leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the same map to sums over the orthogonal idempotents leads to diagonal double Schur function expansions that are identified as τ -functions of hypergeometric type. The second method is the standard construction of τ -functions as vacuum state matrix elements of products of vertex operators in a fermionic Fock space with elements of the abelian group of convolution symmetries. A homomorphism between these two group actions is derived and shown to be intertwined by the characteristic map composed with fermionization. Applications include Okounkov’s generating function for double Hurwitz numbers, which count branched coverings of the Riemann sphere with nonminimal branching at two points, and various analogous combinatorial counting functions.

1

Introduction

Many of the known generating functions for various combinatorial invariants related to Riemann surfaces have been shown to be KP τ -functions, and hence to satisfy the infinite set of Hirota bilinear equations defining the KP hierarchy, or some reduction thereof. These include the Kontsevich matrix integral [19], which is a KdV τ -function, the generator for Hodge invariants [18], the matrix integral that generates single Hurwitz numbers [2], and the one for Belyi curves and dessins d’enfants [1]. Other generating functions are known Work supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Qu´ebec – Nature et technologies (FRQNT). ∗

1

to be τ -functions of the 2D Toda hierarchy, some of which are also representable as matrix integrals. Examples are the Itzykson–Zuber 2-matrix integral, which generates the enumeration of ribbon graphs [16], Okounkov’s generating function for double Hurwitz numbers, counting branched covers of the Riemann sphere with fixed nonminimal branching at a pair of specified points [21], and the Harish-Chandra–Itzykon–Zuber (HCIZ) integral [12, 16], which generates the monotone double Hurwitz numbers [10]. The purpose of this work is to relate two different methods of constructing 2D Toda τ functions [23, 24, 25] as generating functions for geometrical-topological invariants that have combinatorial interpretations involving counting of paths in the symmetric group. These include the double Hurwitz numbers [21], which may be viewed equivalently as counting paths in the Cayley graph from one conjugacy class to another, the monotone double Hurwitz numbers [10], generated by the HCIZ integral in the N → ∞ limit, which count weakly monotone paths, and the mixed double Hurwitz numbers [11], which count a combination of both. To these we add a new family defined by matrix integrals [14] that are variants of the HCIZ integral, which count combinations of weakly monotone and strictly monotone paths. In each case, the generating function can be interpreted as a τ -function of the 2D Toda integrable hierarchy that is of hypergeometric type [13, 14, 22]. The first method is based on combining Frobenius’s characteristic map, from the centre Z(C[Sn ]) of the group algebra C[Sn ] to the algebra Λ of symmetric functions, with automorphisms of Z(C[Sn ]) defined by multiplication by elements of a certain abelian group within Z(C[Sn ]). The second is based on the usual construction of τ -functions [14, 22, 24] as vacuum state matrix elements of products of vertex operators and operators from the Clifford group acting on a fermionic Fock space F . P Under the characteristic map, extended to C[Sn ] ⊗ C[Sn ], the sum g∈Sn n! g ⊗ g over all diagonal elements maps to the diagonal double Schur function expansion given by the Cauchy-Littlewood formula or, equivalently, to a diagonal sum of products of the power sum symmetric functions. This may be interpreted as the restriction of the vacuum 2D Toda τ -function to flow variables given by power sums. Certain homomorphisms of the group algebra, defined by multiplication by central elements consisting of exponentials of linear combinations of power sums in the special set of commuting elements {J1 , J2, J3 , . . .} introduced by Jucys [17] and Murphy [20], give rise to a “twisting” of the expansions in symmetric functions which, depending on the choice of the specific element, produce τ -functions of hypergeometric type [13, 14, 22] that may be interpreted as combinatorial generating functions. The usual way to construct τ -functions of this type is by evaluating the vacuum state matrix elements with a group element that is diagonal in the standard fermionic basis. The abelian group of such diagonal elements is identified as the group Cˆ of convolution symmetries [15]. In Section 2.5 we define a homomorphism I : AP → Cˆ from the group AP of central P∞ elements of the form {e i=0 ti Pi }, where the Pi ’s are the power sums in the Jucys-Murphy elements, to the group Cˆ of convolution symmetries. The elements of AP act on Z(C[Sn ]) 2

by multiplication, and are diagonal in the basis of orthogonal idempotents {Fλ }, labelled by partitions λ corresponding to irreducible representations. The elements of Cˆ act on F and similarly are diagonal in the standard orthonormal basis {|λ; Ni} within each charge N sector FN ⊂ F . Composing the characteristic map with the one defining the Bose–Fermi equivalence [6] gives an injection Fn : Z(C[Sn ]) → F of the centre of the group algebra into the fermionic Fock space that maps the basis of orthogonal idempotents {Fλ } to the orthonormal basis {|λ; 0i}. The main result, stated in Theorem 2.2, is that this map intertwines the action of the group AP on Z(C[Sn ]) with that of Cˆ on F . The action of AP on Z(C[Sn ]), when expressed in another basis {Cλ } consisting of the sums over elements of the conjugacy class with cycle type λ, provides combinatorial coefficients that count paths in the Cayley graph of Sn , starting from an element in the conjugacy class with cycle type λ and ending on one with type µ. These are just the matrix elements of the AP group element in the {Cλ } basis. The image of these basis elements under the characteristic map are, up to normalization, the power sum symmetric functions P Pλ . Applying an element of AP to the diagonal sum g∈Sn n! g ⊗ g introduces a “twist” that is interpretable as a sum over various classes of paths in the Cayley graph. Applying the map ch ⊗ ch to this new element provides a double sum over the power sum symmetric functions Pλ ([x])Pµ ([y]), with coefficients given by the matrix elements of AP which count the number of such paths or, equivalently, a double Schur function expansion of a τ -function of hypergeometric type, corresponding to a specific element of the group Cˆ of convolution symmetries. This can be viewed as a method for constructing identities between double sums over the power sum symmetric functions and diagonal double Schur function expansions without involving the usual sums over irreducible characters of Sn . Several examples of this construction are provided in Section 3, starting with the generating function for the double Hurwitz numbers first studied by Okounkov [21]. In this case, the convolution group element is given by an elliptic θ-function. In the case of the monotone double Hurwitz numbers, it corresponds to convolution with the exponential function. Choosing the expansion parameter that counts the number of steps in a path as z = −1/N, the resulting sequence of 2D Toda τ -functions is just the large N limit of the HCIZ matrix integral [10, 11]. The mixed double Hurwitz numbers are obtained by multiplying the two AP group elements. A fourth class of examples is introduced, in which the generating 2D Toda τ -function is also interpretable as a matrix integral analogous to the HCIZ integral, but with the exponential trace product coupling replaced by a non integer power of the characteristic polynomial of the product [14]. This coupling has also been considered in the study of the spectral statistics of 2-matrix models [3, 4].

3

2 2.1

The characteristic map, twisting homomorphisms and convolution symmetries The characteristic map and the Cauchy-Littlewood formula

Let Λ = C[P1 , P2 , . . .] be the ring of symmetric functions, equipped with the usual projection homomorphism evn,x : Λ → Λn n X Pk 7→ xki

(2.1)

i=1

onto the ring Λn of symmetric polynomials in n variables for each n. The two bases of Λ relevant for our purposes will be the power sum symmetric functions {Pλ := Pλ1 Pλ2 · · · Pλℓ(λ) : λ an integer partition}

(2.2)

and the Schur symmetric functions {Sµ : µ an integer partition},

(2.3)

which are related to each other by the irreducible characters χµ of the symmetric groups as Pλ =

X

χµ (λ)Sµ ,

Sµ =

µ, |µ|=|λ|

X

χµ (λ)

λ, |λ|=|µ|

where, denoting the number of parts of λ equal to i by mi , Y mi ! imi . Zλ :=

Pλ , Zλ

(2.4)

(2.5)

i

The irreducible characters χµ also appear in the change of basis formula between two important bases of the centre Z(C[Sn ]) of the symmetric group algebra, namely the conjugacy class sums, which consist of sums of all permutation with a fixed cycle type λ, ) ( X g : λ an integer partition , (2.6) Cλ := g∈Sn cyc(g)=λ

and the orthogonal idempotents, which correspond to the irreducible representations of Sn , {Fµ : µ an integer partition},

4

(2.7)

and have the useful computational property that Fµ Fµ = Fµ and Fµ Fν = 0 for µ 6= ν. The relation between these bases is given by 1 X 1 X Cλ = Hµ χµ (λ)Fµ , Fµ = χµ (λ)Cλ , (2.8) Zλ Hµ µ, |µ|=|λ|

λ, |λ|=|µ|

where Hµ is the product of the hook lengths of the partition µ, also given by the formulas   χµ (Id) 1 Hµ = . (2.9) = det |µ|! (λi − i + j)! 1≤i,j≤ℓ(λ)

Frobenius’s characteristic map is a linear map that intertwines these changes of bases in Z(C[Sn ]) and Λ, defined by chn : Z(C[Sn ]) → Λ Cλ 7→ Pλ /Zλ

(2.10)

Fµ 7→ Sµ /Hµ . In fact this map is a linear isomorphism if we restrict its codomain to the space of homogeneous symmetric functions of degree n. It will be useful to extend the map chn to the whole group algebra C[Sn ] by defining chn (g) := Pcyc(g) /n!

(2.11)

for a permutation g ∈ Sn . Applying the tensor product map ch ⊗ ch to the element X n! g ⊗ g ∈ C[Sn ] ⊗ C[Sn ]

(2.12)

g∈Sn

then gives

ch ⊗ ch

X

n! g ⊗ g

g∈Sn

!

=

X Y X 1 1 Pλ ([x])Pλ ([y]) = Sµ ([x])Sµ ([y]) = , Zλ 1 − xa yb a,b µ, |µ|=n

λ, |λ|=n

(2.13) where we have identified Λ ⊗ Λ with the ring of symmetric functions in two sets of variables x and y. The last equality is just the Cauchy-Littlewood formula. Restricting the 2D Toda flow variables t = (t1 , t2 , . . .), s = (s1 , s2 , . . .) (2.14) to the power sum values

we have

n

n

1X n y si = i b=1 b

1X n x , ti = i a=1 a n Y

P∞ 1 = e i=1 iti si , 1 − xa yb a,b=1

which is the vacuum 2D Toda τ -function, restricted to the values (2.15). 5

(2.15)

(2.16)

2.2

“Twisting” homomorphisms: multiplication by power sums in the Jucys-Murphy elements

The map (2.13) can be “twisted” by elements of an abelian group AP,n acting on the centre Z(C[Sn ]) to obtain other 2D Toda τ -functions of interest as follows. The Jucys-Murphy elements {Jb ∈ C[Sn ]}b=1,...,n are defined as sums of transpositions, Jb :=

b−1 X

(a b).

(2.17)

a=1

They are easily seen to generate a commutative subalgebra of C[Sn ], and any symmetric polynomial in them is in the centre Z(C[Sn ]). We can adjoin to the ring of symmetric functions Λ a “trivial” element P0 , taking value n under the extended evaluation map evn,J : Λ[P0 ] → Z(C[Sn ]) G 7→ G(J ) Pk 7→ Pk (J ) :=

n X

Jbk

(2.18)

b=1

P0 7→ P0 (J ) := n Id. Remark 2.1. While the trivial element P0 acts like a scalar for any fixed n, it allows us to write down expressions for conjugacy classes Cλ which hold uniformly for all n (see [5, 7]), such as
P2 (J ) − 21 P0 (J ) P0 (J ) − 1 = C31n−3 P1 (J ) = C21n−2 ,
3 1 1 (2.19) 1 = C1n , 2 P11 (J ) − 2 P2 (J ) + 2 P0 (J ) P0 (J ) − 1 = C221n−4 . From these follow equations for products of conjugacy classes such as   n C21n−2 · C21n−2 = 3C31n−3 + 2C221n−4 + C1n . 2

(2.20)

In this way, the ring Λ[P0 ] can be seen as an inverse limit of the centres Z(C[Sn ]) for all n ∈ N, sometimes called the Farahat-Higman algebra [8].

Endomorphisms of Z(C[Sn ]) consisting of multiplication by a central element are diagonal in the basis {Fµ } of orthogonal idempotents. For elements of the form G(J ), the result of Jucys [17] and Murphy [20] gives the eigenvalues as
G(J )Fµ = G cont(µ) Fµ , (2.21)

where cont(µ) is the multi set (possibly with repeated values) of contents of the boxes (i, j) appearing in the Young diagram for the partition µ, cont(µ) := {j − i : (i, j) ∈ µ}. 6

(2.22)

Our “twisting” of the map ch ⊗ ch is defined to act on the second tensor factor only through multiplication by a symmetric function G(J ) for G ∈ Λ[P0 ] before applying the Frobenius characteristic map:
ch ⊗ ch ◦ G(J ) : Z(C[Sn ]) ⊗ Z(C[Sn ]) → Λ ⊗ Λ. (2.23)

Using (2.21), it is easy to compute the result of applying the twisted homomorphism (2.23) to the element (2.12) in the basis {Sλ ([x])Sµ ([y])} in three steps. First we apply the map ch to the left tensor factor X X n! g ⊗ g 7→ Hµ Sµ ([x]) ⊗ Fµ . (2.24) ch ⊗Id : g∈Sn

µ, |µ|=n

Then we multiply the right tensor factor of multiplication by G(J ) X X
Id ⊗ G(J ) : Hµ Sµ ([x]) ⊗ Fµ 7→ G cont(µ) Hµ Sµ ([x]) ⊗ Fµ . µ, |µ|=n

(2.25)

µ, |µ|=n

And finally, we apply the map ch to the right tensor factor X X

Id ⊗ ch : G cont(µ) Hµ Sµ ([x]) ⊗ Fµ 7→ G cont(µ) Sµ ([x])Sµ ([y]). µ, |µ|=n

(2.26)

µ, |µ|=n

As will be seen in Section 2.4, this is the restriction of a 2-KP τ -function of hypergeometric type to the values (2.15) of the flow parameters which, by suitable normalization, can be extended to a Z-lattice of 2D Toda τ -functions. We can perform the same computation in the basis {Pλ ([x])Pµ ([y])} instead. Multiplying the basis elements Cλ by G(J ) gives a linear combination X G(J )Cλ = Gλµ Cµ , (2.27) µ

where the coefficients Gλµ are given in general by the character sum
1 X Gλµ = G cont(ν) χν (λ)χν (µ). Zλ ν

(2.28)

As will be seen below, in many cases Gλµ is a combinatorial number, counting certain types of paths in the Cayley graph of Sn from an element in the conjugacy class of type Cλ to one in the class Cµ . Applying the twisted homomorphism (2.23) to the element (2.12) in three steps again gives X X n! g ⊗ g 7→ Pλ ([x]) ⊗ Cλ (2.29) ch ⊗Id : g∈Sn

Id ⊗ G(J ) :

X

Pλ ([x]) ⊗ Cλ 7→

λ, |λ|=n

Id ⊗ ch :

X

λ, |λ|=n

X

Gλµ Pλ ([x]) ⊗ Cµ

(2.30)

λ, µ, |λ|=|µ|=n

Gλµ Pλ ([x]) ⊗ Cµ 7→

λ, µ, |λ|=|µ|=n

X

Gλµ Pλ ([x])Pµ ([y]).

λ, µ, |λ|=|µ|=n

7

(2.31)

Comparing (2.26) and (2.31), we get a twisted version of (2.13): ! X X X

Gλµ Pλ ([x])Pµ ([y]) = G cont(µ) Sµ ([x])Sµ ([y]). = n! g ⊗ G(J )g ch ⊗ ch λ, µ |λ|=|µ|=n

g∈Sn

µ, |µ|=n

(2.32)

2.3

Interpretation as generating functions

We now consider the combinatorial meaning of the coefficients Gλµ . If the operator G(J ) is taken to be the power series in a formal parameter z given by P1 (J )z

Gc (z, J ) := e

=

∞ X

P1 (J )k

k=0

zk , k!

(2.33)

then the coefficient of z k /k! in Gc (z, J ) is the element P1 (J )k = C1kn .

(2.34)

This acts on the group algebra C[Sn ] by multiplication by every possible product (a1 b1 )(a2 b2 ) · · · (ak bk )

(2.35)

of k (not necessarily disjoint, nor even distinct) transpositions. Thus, for any pair of permutations g, h ∈ Sn , the coefficient of g ⊗ h z k /k! in the element X
n! g ⊗ Gc (z, J )g (2.36) g∈Sn

is the number of solutions in Sn of the equation h = (a1 b1 )(a2 b2 ) · · · (ak bk )g,

(2.37)

which is precisely the number of k-step walks from the vertex g to the vertex h in the Cayley graph of Sn generated by all transpositions. If we then apply the characteristic map ch ⊗ ch to this element, as in (2.32), we see that the coefficient Gλµ in this case is the generating function for k-step walks in the Cayley graph from any vertex g with cycle type λ to any vertex h with cycle type µ. As another example, take the operator G(J ) to be the power series H(z, J ) :=

n Y b=1

∞

X 1 = zk 1 − zJb k=0 b

X

1 ≤b2 ≤···≤bk

8

Jb1 Jb2 · · · Jbk .

(2.38)

Then the coefficient of z k in H(z, J ) is the operator on C[Sn ] which acts by multiplication by every possible product (2.35) subject to the restriction that b1 ≤ b2 ≤ · · · ≤ bk ,

(2.39)

where ai < bi by convention. The corresponding walks in the Cayley graph are called (weakly) monotone walks, and for this choice of operator G(J ), the coefficient Gλµ in (2.32) is the generating function for k-step weakly monotone walks in the Cayley graph from any permutation with cycle type λ to any permutation with cycle type µ. These are precisely the (nonconnected) monotone double Hurwitz numbers [10]. As a final example of an operator G(J ) with a combinatorial meaning, we can take n ∞ Y X E(z, J ) := (1 + zJb ) = zk b=1

k=0

X

Jb1 Jb2 · · · Jbk ,

(2.40)

b1 0, if j = 0, if j < 0,

(2.79)

(2.80)

(2.81)

It follows that [z] Cˆρ([z]) |λ; 0i = eθ0 |λ| rλ (0)|λ; 0i, [z]

(2.82)

where rλ (0) is defined in (2.68). We then have the following: L Theorem 2.2. The map F : n≥0 Z(C[Sn ]) → F0 intertwines the multiplicative action L ˆ of the group AP on n≥0 Z(C[Sn ]) with the linear action of the group C on F0 via the ˆ homomorphism I : AP → C. 13

Proof. This follows from the fact that, up to scaling, the linear map F takes Fλ into |λ; 0i and these are,Prespectively, eigenvectors of the automorphism of Z(C[Sn ]) defined by multiP∞
∞ t P t P plication by e i=0 i i and I e i=0 i i which, as given by (2.69) and (2.82), have the same [z] eigenvalue et0 |λ| rλ (0). Remark 2.3. Note that on the intermediate space Λ of the composition F:

M

Z(C[Sn ]) → Λ → F0 ,

(2.83)

n≥0

multiplication by P0 (J ) ∈ AP corresponds to the Eulerian operator ∞ X

∞

kPk

X ∂ ∂ xi = , ∂Pk ∂xi

(2.84)

i=1

k=1

while multiplication by P1 (J ) ∈ AP corresponds to the cut-and-join operator of [9, 10, 11, 18],  ∞  1 X ∂2 ∂ + ijPi+j . (i + j)Pi Pj 2 ∂Pi+j ∂Pi ∂Pj

(2.85)

i,j=1

Remark 2.4. Alternatively, the homomorphism I : AP → Cˆ may be defined by P∞

e

i=0 θi Pi

P∞

P∞

7→ e

Pj

k=1 k

i=0 θi

j∈Z

i




:ψj ψj† :

(2.86)

,

P where the sum jk=1 ki is defined for j ≤ 0 by interpreting it as a polynomial in j of degree i + 1. ˆ to the operator Thus, multiplication by Pi (J ) ∈ AP corresponds, on C, j ∞ X X j∈Z

3 3.1

k=1


ki :ψj ψj† :.

(2.87)

Examples Double Hurwitz numbers

Following Okounkov [21], for a pair of parameters (β, q), we choose β

ρj = q j e 2 j(j+1) ,

rj = qejβ ,

1

β

ρ˜j = ρj q 2 e 8 .

(3.1)

(The choice ρ˜j is used in [21]; the choice ρj fits more naturally with the conventions of Theorem 2.2. For N = 0, which is the only case needed, the two τ -functions coincide. The relationship between the two for general N is indicated below.) It follows that 1

β

rλ (N) = q 2 N (N −1) e 6 N (N

14

2 −1)

q |λ| eβN |λ| eβ contλ ,

(3.2)

where

ℓ(λ) X
1X := λi (λi − 2i + 1), P contλ (j − i) = 1 cont(λ) = 2 i=1

(3.3)

(i,j)∈λ

or

N

r˜λ (N) = rλ (N)q 2 e

βN 8

.

(3.4)

Defining 1X k j :ψj ψj† : for k ∈ N+ , Fˆk := k j∈Z

ˆ := N

X

:ψj ψj† :,

j∈Z

the convolution symmetry elements corresponding to ρ and ρ˜ are 1 ˆ β ˆ Cˆρ˜ = Cˆρ q 2 N e 8 N .

ˆ ˆ Cˆρ = eln q F1 +β F2 ,

(3.5)

(3.6)

The Fourier coefficients of the element ρ(z) of L2 (S 1 ) are thus given by β 2

ρj = ej ln q+ 2 j

Summing, we obtain an elliptic θ-function   β β X X e z β e2 j ln q+ β2 j 2 j −j−1 β2 (j+1)2 . θ ; ej e = z q e = q q 2 j∈Z j∈Z

(3.7)

(3.8)

Under the homomorphism (2.78), the element of AP mapping to Cˆρ is thus I : eln q P0 +βP1 7→ Cˆρ .

(3.9)

The corresponding 2D Toda τ -functions are Toda (N, t, s) = hN|ˆ γ+ (t)Cˆρ γˆ (s)|Ni τC2D ρ

(3.10)

N βN Toda Toda (N, t, s). τC2D (N, t, s) = hN|ˆ γ+ (t)Cˆρ˜γˆ (s)|Ni = q 2 e 8 τC2D ρ ρ ˜

(3.11)

The generating function for the double Hurwitz numbers [21] ∞ ∞ X X βb X n Hurb (λ, µ) Pλ(t)Pµ(s), q FCρ (t, s) = b! λ, µ n=1 b=0

(3.12)

|λ|=|µ|=n

which counts only simply connected branched coverings of CP1 , is then the logarithm   2D Toda (0, t, s) (3.13) FCρ (t, s) = ln τCρ of

Toda (0, t, s) = τC2D ρ

∞ X n=1

qn

∞ X βb X b=0

b!

Covb (λ, µ) Pλ(t)Pµ (s),

(3.14)

λ, µ |λ|=|µ|=n

where n is the number of sheets in the covering, b is the number of simple branch points in the base, λ and µ are the ramification types at 0 and ∞, and Covb (λ, µ) is the total number of such coverings. 15

3.2

Monotone double Hurwitz numbers

Consider the Harish-Chandra–Itzykson–Zuber (HCIZ) integral
! Z −zN ai bj N −1 Y det e † 1≤i,j≤N IN (z, A, B) = e−zN tr(U AU B) dµ(U) = k! ∆(a)∆(b) U ∈U (N ) k=0

(3.15)

where dµ(U) is the Haar measure on U(N), A and B are a pair of diagonal matrices with eigenvalues a = (a1 , . . . , aN ), b = (b1 , . . . , bN ) respectively, and ∆(a), ∆(b) are the Vandermonde determinants. Defining
1 ln I (z, A, B) , N N2

FN :=

(3.16)

it was shown in [10] that this admits an expansion FN =

∞ ∞ X X (−z)n X g=0 n=0

n!

~ g (λ, µ) Pλ ([A])Pµ ([B]) , H (−N)2g+ℓ(λ)+ℓ(µ)

(3.17)

λ, µ |λ|=|µ|=n ℓ(λ), ℓ(µ)≤N

~ g (λ, µ) is a monotone double Hurwitz number, which equals the number of transitive where H r-step monotone walks in the Cayley graph of Sn from a permutation with cycle type λ to one with cycle type µ and r = 2g − 2 + ℓ(λ) + ℓ(µ). (3.18) It is also well-known that the HCIZ integral IN (z, A, B) is, within the normalization factor 1 r0exp (N) := QN −1 k=0

k!

,

(3.19)

equal to the 2D Toda τ -function τ HCIZ (N, t, s) with double Schur function expansion [15] X exp r0exp (N)IN (z, A, B) = τ HCIZ (N, t, s) := rλ (N)Sλ (t)Sλ (s), (3.20) λ ℓ(λ)≤N

where (−zN)|λ|  , rλexp (N) = Q N −1 k! N (λ) k=0

(N)λ =

Y

(N + j − i),

(3.21)

(i,j)∈λ

evaluated at the parameter values

t = [A],

s = [B]. 16

(3.22)

This may be expressed as the fermionic vacuum state expectation value γ+ ([A])Cˆexp γˆ− ([B])|Ni = h0|ˆ γ+ ([A])Cˆexp (N)ˆ γ− ([B])|0i τCexp (N, [A], [B]) = hN|ˆ

(3.23)

where P∞

Cˆexp := e

j=0 (j

ln(−zN )−ln(j!)):ψj ψj† :

P∞

† j=−N ((j+N ) ln(−zN )−ln((j+N )!)):ψj ψj :

Cˆexp (N) := e

(3.24) .

(3.25)

ˆ The convolution group element Cˆexp (N) is the image, under the homomorphism I : AP → C, of the element P∞

(−1/N)i

e− ln(−zN )P0 (J ) e i=1 i Pi (J ) = e− ln(−zN )P0 (J ) H(−1/N, J )   − ln(−zN )P0 (J ) ln q P0 (J )+βP1 (J ) = Cˆexp (N). I e H(−1/N, J )e

3.3

(3.26) (3.27)

Mixed double Hurwitz numbers

The mixed monotone Hurwitz numbers are defined in [11] as the number of r-step walks in the Cayley graph of Sn from a permutation with cycle type λ to one with cycle type µ, subject to the restriction that the first p ≤ r steps form a weakly monotone walk, and the last r − p steps are unrestricted. This case has a generating function that is obtained by composing the group element (3.9) in AP corresponding to the ordinary double Hurwitz numbers with the one (3.26) corresponding to the monotone ones. Applying the homomorphism AP to the product therefore gives the product of the convolution group elements   ˆ ˆ ln q P0 (J )+βP1 (J ) − ln(−zN )P0 (J ) I e e H(−1/N, J ) = eln q F1 +β F2 Cˆexp (N). (3.28) It follows that the factor rλ (N) that enters in the double Schur function expansion of the corresponding mixed double Hurwitz number generating function is given by the product of the ones for these two cases, 1

β

rλ (N) = q 2 N (N −1) e 6 N (N

3.4

2 −1)

(−zN)|λ|  . q |λ| eβN |λ| eβ contλ Q N −1 k! N (λ) k=0

(3.29)

Determinantal matrix integrals as generating functions

Following [14, Appendix A], we can obtain a new class of combinatorial generating functions that generalize the case of the HCIZ integral as follows. Choose a pair (α, q) of (real or complex) parameters, with α not a positive integer, and define ( j q (1−α)j if j ≥ 1, (α,q) j! := ρj (3.30) 1 if j ≤ 0, 17

where (a)j := a(a + 1) · · · (a + j − 1)

(3.31)

is the (rising) Pochhammer symbol. Then (α,q)

(α,q)

rj and (α,q)

Tj

:=

ρj

(α,q)

ρj−1

(

=

q(j−α) j

if j ≥ 1, if j ≤ 0,

1

(3.32)

( j ln q + ln(1 − α)j − ln(j!) if j ≥ 1, = 0 if j ≤ 0.

:= ln ρ(α,q) j

For any N ∈ N, let

P∞

Cˆ(α,q) (N) := e

(α,q)

j=−N

Tj+N :ψj ψj† :

(3.33)

(3.34)

be the corresponding shifted convolution symmetry group element. We then have, for ℓ(λ) ≤ N, (α,q) Cˆ(α,q) (N)|λ; 0i = rλ (N)|λ; 0i

(3.35)

where (α,q)

rλ

(α,q)

(N) = r0

(N)

Y

(α,q)

(α,q)

rN +j−i = r0

(N)q |λ|

(i,j)∈λ

with (α,q) r0 (N)

:=

N −1 Y

(α,q) ρj

=q

1 N (N −1) 2

j=0

j=0

and

N −1 Y

(N − α)λ , (N)λ

(1 − α)j j!

ℓ(λ) Y (a)λ := (a − i + 1)λi

(3.36)

(3.37)

(3.38)

i=1

the extended Pochhammer symbol corresponding to the partition λ = (λ1 , . . . , λℓ(λ) ). For N ∈ N+ , we have the 2D Toda chain of τ -functions X (α,q) τC(α,q) (N, t, s) = h0|ˆ γ+ (t)Cˆ(α,q) (N)γ− (s)|0i = rλ (N)Sλ (t)Sλ(s), (3.39) λ

evaluated at the parameter values (3.22). As shown in [14], this is just the matrix integral Z (α,q) τC(α,q) (N, [A], [B]) = r0 (N) det(IN − qUAU † B)N −α dµ(U) (3.40) U ∈U (N )

=

det(1 − qai bj )1≤i,j≤N ∆(a)∆(b) 18


α−1

.

(3.41)

We now use the other construction to derive an interpretation of this as a combinatorial generating function. Evaluating the generating function for the elementary symmetric polynomials at the Jucys-Murphy elements n Y E(w, J ) := (1 + wJa )

(3.42)

a=1

defines an element of AP . Applying the product  qz P0 − H(z, J )E(w, J ) w to the orthogonal idempotent Fλ , we obtain  qz P0 (1/w)λ H(z, J )E(w, J )Fλ = q |λ| − Fλ . w (−1/z)λ

(3.43)

(3.44)

Specializing to the values

1 , t = [A], s = [B] (3.45) N −α and choosing ℓ(λ) ≤ N, we obtain the same eigenvalue, within a normalization factor, as in (3.35), namely (α,q)  α P0 rλ (N) |λ| (N − α)λ q Fλ = (α,q) −1 H(−1/N, J )E(−1/(N − α), J )Fλ = q Fλ . N (N)λ r0 (N) (3.46) ˆ Under the homomorphism I : AP → C, we thus have P0  α Cˆ(α,q) (N) −1 H(−1/N, J )E(−1/(N − α), J ) 7→ (α,q) . (3.47) q N r0 (N) z = −1/N,

w=

Applying the product q P0 H(z, J )E(w, J ) to the conjugacy class sum Cλ therefore gives P0

q H(z, J )E(w, J )Cλ =

∞ X

zk wl

X

q |λ| Ek,l (λ, µ)Cµ,

(3.48)

µ

k, l=0

where, similarly to the mixed double Hurwitz numbers, Ek,l (λ, µ) is the number of (k+l)-step walks in the Cayley graph of Sn starting at a permutation with cycle type λ and ending at a permutation of cycle type µ which obey the condition that the first k steps form a weakly monotone walk, and the last l steps form a strictly monotone walk.
P Applying the map ch ⊗ ch to g∈Sn n! g ⊗ q P0 H(z, J )E(w, J )g gives X

g∈Sn

P0




n! g ⊗ q H(z, J )E(w, J )g 7→

∞ X

zk wl

k, l=0

=

X

|λ|=n

19

X

q |λ| Ek,l (λ, µ)Pλ(t)Pµ (s)

|λ|=|µ|=n |λ|

q rλ (z, w)Sλ (t)Sλ (s),

(3.49)

where rλ (z, w) :=

Y 1 + (j − i)w 1 − (j − i)z

(3.50)

(i,j)∈λ

and hence

 (α,q)  α |λ|  1 rλ (N) 1 q = (α,q) −1 rλ − , . N N N −α r0 (N)

(3.51)

Therefore, in the limit N → ∞, the matrix integral (3.40) is the generating function for the number of weakly-monotonic-then-strictly-monotonic double Hurwitz numbers.

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[24] T. Takebe, “Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy I”, Lett. Math. Phys. 21 77–84, (1991). 2 [25] K. Ueno and K. Takasaki, “Toda Lattice Hierarchy”, in:Group Representation and Systems of Differential Equations, 1–95, Adv. Stud. in Pure Math. 4, (1984). 2 N.B. The numbers at the end of each entry indicate the pages where the reference is cited.

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