On some polynomials enumerating Fully Packed Loop configurations: Evaluation at negative values

Advances in Applied Mathematics 51 (2013) 446–466

Contents lists available at SciVerse ScienceDirect

Advances in Applied Mathematics www.elsevier.com/locate/yaama

On some polynomials enumerating Fully Packed Loop configurations: Evaluation at negative values Tiago Fonseca Centre de Recherches Mathématiques, Université de Montréal, Canada

a r t i c l e

i n f o

Article history: Received 6 March 2013 Revised 29 May 2013 Accepted 31 May 2013 Available online 20 June 2013 MSC: primary 05A15 secondary 12D05, 82B23 Keywords: Fully Packed Loop O(n) Loop model Integrable systems Razumov–Stroganov conjecture Knizhnik–Zamolodchikov equation

a b s t r a c t In this article, we are interested in the enumeration of Fully Packed Loop configurations on a grid with a given noncrossing matching. These quantities also appear as the groundstate components of the O (n) Loop model as conjectured by Razumov and Stroganov and recently proved by Cantini and Sportiello. When considering matchings with p nested arches these quantities are known to be polynomials. In a recent article, Fonseca and Nadeau conjectured some unexpected properties of these polynomials, suggesting that these quantities could be combinatorially interpreted even for negative p. Here, we prove some conjectures in this article. Notably, we prove that for negative p we can factor the polynomials into two parts a “positive” one and a “negative” one. Also, a sum rule of the negative part is proven here. © 2013 Elsevier Inc. All rights reserved.

0. Introduction In 2001, Razumov and Stroganov [15] conjectured that there is a correspondence between the Fully Packed Loop (FPL) configurations, a combinatorial model, and the components of the groundstate vector in the O (n) Loop model, a model in statistical physics. On the one hand, the connectivity of the FPL configurations at the boundary is described by a perfect noncrossing matchings π of 2n points. The number of FPL configurations associated to a certain matching π is denoted by A π . On the other hand, the O (n) model is defined on the set of matchings and the groundstate components are naturally indexed by the matchings and are written Ψπ . Razumov and Stroganov conjectured that this quantities are the same A π = Ψπ for all matchings π . This conjecture was proved in 2010 by Cantini and Sportiello [2].

E-mail address: [email protected]. 0196-8858/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.aam.2013.05.003

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

Consider matchings with p nested arches surrounding a smaller matching

447

π , which we denote by

(π ) p = (· · · (π ) · · ·). It was conjectured in [19], and after proved in [3,12], that the quantities A (π ) p and Ψ(π ) p are polynomials in p. In a recent article, Nadeau and Fonseca [10] conjectured some surprising properties of these polynomials. The goal of this article is to prove some of these conjectures, notably Conjectures 3.8 and 3.11. Let π be a matching composed by n arches. We denote by A π ( p ) (respectively Ψπ ( p )) the polynomial which coincides with A (π ) p (respectively Ψ(π ) p ) when p is a nonnegative integer. In this paper we prove that, for p between 0 and −n, these quantities are either zero or they can be seen as the product of two distinct terms. One being a new quantity G π also indexed by perfect noncrossing matchings and the other being again the quantities A π . The quantities G π are surprisingly connected with the Fully Packed Loop model: the sum of the absolute values of G π is equal to the number of FPL configurations. This relation has been proven in [9]. Here we prove another sum rule also conjectured in [10]: the sum of the quantities G π is equal to the number of vertically symmetric FPL configurations, up to an eventual sign. These sum rules, together with other properties of G π , rises the idea that these numbers G π have some combinatorial meaning, i.e. they are counting something which is related to the FPL configurations. An interesting byproduct of the proofs are the multivariate integral formulæ proposed for G π , which allow us to reformulate the first mentioned conjecture in a stronger form (a polynomial form). Let us give a detailed outline of this article. In the first section we state the two conjectures that we solve here. They concern the polynomials A π (t ). In order to prove the first one, we introduce a multivariate polynomial version of the quantities Ψπ in Section 2, defined though the quantum Knizhnik–Zamolodchikov equation. Although it seems a more complicated approach, this version allows us to use some polynomial properties, which will be essential to the proof. The two further sections are dedicated to the proof of the conjectures. The paper finishes with Appendix A, where we describe some important results, which are straightforward but a little bit tedious. 1. Conjectures The aim of this article is to prove two conjectures presented in [10], and therefore we use the same notations and the reader should find all necessary definitions in Sections 1 and 2 of that paper. These conjectures are about the polynomials Ψπ (t ) for negative t. When computing these quantities in the context of the O (n) Loop model, it is natural to add an extra parameter τ , i.e. there is a bivariate polynomial Ψπ (τ , t ) which has the same properties as Ψπ (t ) and in the limit τ = 1, it coincides with Ψπ (t ). In Section 2, where we explain how to compute the Ψπ (t ), the origin of this parameter will be made more clear. For now, it will be enough to think at this parameter as a refinement. 1.1. Integer roots Let π be a matching, represented by a link pattern, and |π | = n its number of arches. Define xˆ := 2n + 1 − x. Definition 1.1 (m p (π )). Let p be an integer between 1 and n − 1. We consider the set A Lp (π ) of arches

{a1 , a2 } such that a1  p and p < a2 < pˆ , and the set A Rp (π ) of arches {a1 , a2 } such that p < a1 < pˆ and a2  pˆ . It is clear that |A Lp (π )| + |A Rp (π )| is a even nonnegative integer, and we can thus define the nonnegative integer

m p (π ) :=

|A Lp (π )| + |A Rp (π )| 2

.

448

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

For example, let

π be the following matching with eight arches. For p = 4, we get |ALp (π )| = 3

and |A Rp (π )| = 1, which count arches between the regions (O) and (I), thus m4 (π ) = 2. In the figure on the right we give an alternative representative by folding the link pattern, it is then clear that m p (π ) is half of the number of arches linking (O) with (I).

The reader can check that m p = 0, 1, 2, 2, 2, 1, 1 for p = 1, . . . , 7. It was conjectured in [10] that these numbers correspond to the multiplicity of the real roots of Ψπ (t ): Conjecture 1.2. All the real roots of the polynomials Ψπ (t ) are negative integers, and − p appears with multiplicity m p (π ). Equivalently, we have a factorization:

Ψπ (t ) =

1

|d(π )|!

|π |−1   m p (π ) (t + p ) · · Q π (t ), p =1

where Q π (t ) is a polynomial with integer coefficients and no real roots. This conjecture seems to hold in the case of the bivariate polynomial Ψπ (τ , t ), the only difference being that Q π depends now on τ . In Section 3, we prove a weaker version of this conjecture: Ψπ (τ , − p ) = 0 if m p (π ) = 0. 1.2. Values at negative p We are now interested in the value of the polynomial Ψπ (τ , − p ) for integer values 0  p  n. It has been already conjectured that it vanishes if m p (π ) = 0. So let π be a matching and p such that m p (π ) vanishes. It means that there are no arches that link the outer part with the inner part of π . I.e. we can define a matching sitting in the outer part (denote it by α ) and the other in the inner part (denote it by β ), as shown in the picture:

π= We introduce the notation

π = α ◦ β to describe this situation. We need one more definition:

Definition 1.3 (G π ). For any matching

π we define G π (τ ) := Ψπ





τ , −|π | ,

and G π := G π (1). We are now ready to present the main result of this article: Theorem 1.4 (Generalization of Conjecture 3.8 of [10]). Let π be a matching and p be an integer between 1 and |π | − 1 such that m p (π ) = 0, and write π = α ◦ β with |α | = p. We then have the following factorization:

Ψπ (τ , − p ) = G α (τ )Ψβ (τ ).

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

449

Notice that we are reducing the number of unknowns from cn to c p + cn− p − 1. The proof is postponed to Section 3. 1.3. Sum rules It has been proved in [9] that these numbers G π (τ ) have some interesting properties. For example G π (−τ ) = (−1)d(π ) G π (τ ) and Theorem 1.5. (See [9].) We have the sum rule:



G π (τ ) =

π : |π |=n



Ψπ (−τ ).

π : |π |=n

This can be used to partially prove Conjecture 3.11 of [10]. Notice that, according to Conjecture 1.2,

(−1)d(π ) G π = |G π |. Theorem 1.6. For any positive integer n, we have



(−1)d(π ) G π = An ,

(1)

π : |π |=n



G π = (−1)

n(n−1) 2



A nV

2

,

(2)

π : |π |=n

where d(π ) is the number of boxes of the Young diagram Y (π ). Eq. (1) hasbeen proved  in [9], and it follows from Theorem 1.5. Here we will prove Eq. (2). The point is that π G π = π Ψπ (−1) is equivalent to the minus enumeration of TSSCPP which appears in Di Francesco’s article [5], and this can be computed, see Section 4 for a better explanation. 2. Multivariate solutions of the O (n) Loop model In this section, we briefly describe a multivariate version of the O (n) Loop model. This version, although more complicated, is useful to the proof of Theorem 1.4. ˇ The O (n) model is an integrable model, meaning that there is an operator called R-Matrix which obeys to the Yang–Baxter equation. We can add new parameters { z1 , z2 , . . . , z2n }, called spectral parameters, which will characterize each column. In this multivariate setting, the groundstate depends on the 2n spectral parameters (and in an extra parameter q), in fact the components of the groundstate can be normalized such that they are homogeneous polynomials ψπ ( z1 , . . . , z2n ) of degree n(n − 1). The important fact about these solutions is that we recover the solutions of the O (n) model in the limit zi = 1 for all i, and q = e 2π i /3 . We shall not describe this model in detail here, such detailed description can be found in [12,18, 7,9]. In order to simplify notation we will use z = { z1 , . . . , z2n }, thus we will often write ψπ ( z). Notice that these polynomials depend on q, but we omit this dependence. 2.1. The quantum Knizhnik–Zamolodchikov equation The groundstate of the multivariate O (n) model is known to solve the quantum Knizhnik– Zamolodchikov (qKZ) equation in the special value q = e 2π i /3 . See a complete explanation in [18,9].

450

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

The qKZ equation was firstly introduced in a paper by Frenkel and Reshetkhin [13]. Here we use ˇ be the following operator, the version introduced by Smirnov [16]. Let the R-Matrix

Rˇ i ( zi , zi +1 ) =

qzi +1 − q−1 zi

qzi − q−1 zi +1

Id +

z i +1 − z i

qzi − q−1 zi +1

ei .

The quantum Knizhnik–Zamolodchikov equation:

• The exchange equation: Rˇ i ( zi , zi +1 )ψ( z1 , . . . , zi , zi +1 , . . . , z2n ) = ψ( z1 , . . . , zi +1 , zi . . . . , z2n ),

(3)

for i = 1, . . . , 2n. • The rotation equation:

ρ −1 ψ(z1 , z2 , . . . , z2n ) = κ ψ(z2 , . . . , z2n , sz1 ), where

(4)

κ is a constant such that ρ −2n = 1. In our case s = q6 and κ = q3(n−1) .

2.2. Solutions of the quantum Knizhnik–Zamolodchikov equation We start for pointing down some properties of the solutions of the qKZ equation without proof.

• The solutions are homogeneous polynomials in 2n variables; • The total degree is n(n − 1) and the individual degree in each zi is n − 1; • They obey to the wheel condition:



P ( z1 , . . . , z2n ) z =q2 z =q4 z = 0 ∀k > j > i . j i k In fact these three properties define a vector space: Definition 2.1 (Vn ). We define Vn as the vector space of all homogeneous polynomials in 2n variables, with total degree δ = n(n − 1) and individual degree δi = n − 1 in each variable which obey to the wheel condition. This vector space has dimension cn , exactly the number of matchings of size |π | = n. Moreover, the polynomials ψπ ( z) verify the following important lemma: Lemma 2.2. (See [8].) Let q = {q1 , . . . , q2n }, where i = ±1 are such that changing q−1 into “(” and changing q into “)” gives a valid parenthesis word π ( ). Then

  ψπ q = τ d(π ) δπ , , where δπ , = 1 when we have π ( ) = π . τ is related with q by the formula τ = −q − q−1 . Since there are cn polynomials ψπ ( z), this lemma shows that these polynomials form a basis of Vn . Thus a polynomial in this space is determined by its value on these points q .

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

451

2.3. A different approach We now define another set of polynomials, introduced in [8], φa ( z1 , . . . , z2n ) (indexed by the increasing sequences defined earlier), by the integral formula:



φa ( z) = kn



qzi − q−1 z j

1i < j 2n

×

···

 n dw i i =1

2π i





− w i )(qw i − q−1 w j )  , −1 z ) k 1kai ( w i − zk ) ai


(5)

where the integral is performed around the zi but not around q−2 zi , and kn = (q − q−1 )−n(n−1) . It is relatively easy to check that these polynomials belong to the vector space Vn , so we can write:

φa ( z) =



C a,π (τ )ψπ ( z),

π

where C a,π (τ ) are the coefficients given by the formula:

  φa q = τ d( ) C a, (τ ). An algorithm to compute the coefficients C a,π (τ ) is given in [8, Appendix A]. We just need the following facts: Proposition 2.3. (See [12, Lemma 3].) Let a and π be two matchings. Then we have:

C a,π (τ ) =

⎧ ⎨0 1 ⎩ P a,π (τ )

if π  a; if π = a; if π ≺ a,

where P a,π (τ ) is a polynomial in τ with degree  d(a) − d(π ) − 2. Moreover, we have

C a,π (τ ) = (−1)d(a)−d(π ) C a,π (−τ ),

(6)

since it is a product of polynomials U s in τ with degree of the form d(a) − d(π ) − 2k, k ∈ N, and parity given by d(a) − d(π ): this is an easy consequence of [12, p. 12 and Appendix C]. By abuse of notation, we write (a) p to represent {1, . . . , p , p + a1 , . . . , p + an }, since this corresponds indeed to adding p nested arches to π (a) via known bijections (see [10]). Then one easy but important lemma for us is the following: Lemma 2.4. (See [12, Lemma 4].) The coefficients C a,π (τ ) are stable, that is:

C (a) p ,(π ) p (τ ) = C a,π (τ ) ∀ p ∈ N. 1 We remark that Proposition 2.3, Eq. (6) and Lemma 2.4 also hold for the coefficients C a−,π (τ ) of the inverse matrix.

452

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

2.4. The homogeneous limit The bivariate polynomials Ψπ (τ , t ) are defined as the homogeneous limit of the previous multivariate polynomials (i.e. zi = 1 for all i). Though we are mostly interested in the case τ = 1, since we recover the groundstate Ψπ (t ) = Ψπ (1, t ), as explained in [18], the variable τ it will be useful for the proofs presented here. Define Φa (τ ) := φa (1, . . . , 1).1 Using variable transformation

wi − 1

ui =

qw i − q−1

,

we obtain the formula:

Φa (τ ) =

···

 du i 2π

i



(u j a iu i i j >i

− u i )(1 + τ u j + u i u j ).

(7)

Thus, we can then obtain the Ψπ (τ ) via the matrix C (τ ):

Φa (τ ) =



C a,π (τ )Ψπ (τ );

(8)

π



Ψπ (τ ) =

−1 Cπ ,a (τ )Φa (τ ).

(9)

a

Let aˆ i be the components of (a) p . Now

Φ(a) p (τ ) =

···

n +p i

=

···

aˆ

2π iu i i j >i

 n du i i

 (u j − u i )(1 + τ u j + u i u j )

du i

2π

a iu i i

(1 + τ u i ) p

 (u j − u i )(1 + τ u j + u i u j ), j >i

where we integrated in the first p variables and renamed the rest u p +i → u i . This is a polynomial in p, and we will naturally note Φa (τ , t ) the polynomial such that Φa (τ , p ) = Φ(a) p (τ ). Finally, from Eq. (9) and Lemma 2.4 we obtain the fundamental equation

Ψπ (τ , t ) =



−1 Cπ ,a (τ )Φa (τ , t ).

a

In the special case

τ = 1, we write C a,π = C a,π (1), Φa (t ) = Φa (1, t ) and thus A π (t ) = Ψπ (t ) =



−1 Cπ ,a Φa (t ),

a

thanks to the Razumov–Stroganov conjecture.

1

Notice that φa ( z) depends on q, even if we do not write it explicitly.

(10)

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

453

3. Decomposition formula The aim of this section is to prove Theorem 1.4. With this in mind, we introduce two new multivariate polynomials ψπ ,− p ( z) and g π ( z), which generalize the quantities Ψπ (τ , − p ) and G π (τ ) respectively. 3.1. New polynomials Let π be a matching with size |π | = n and p be a nonnegative integer less or equal than n. We require that the new object ψπ ,− p ( z) have the following essential properties:

• It generalizes Ψπ (τ , − p ), i.e. Ψπ (τ , − p ) = ψπ ,− p (1, . . . , 1); • When p = 0, we have ψπ ,0 (z) = ψπ (z), justifying the use of the same letter; • They are polynomials on zi . Thus, we can define a multivariate version of G π (τ ), by

g π ( z1 , . . . , z2n ) := ψπ ,−|π | ( z1 , . . . , z2n ),

(11)

such that G π (τ ) = g π (1, . . . , 1). Surprisingly enough, using these new polynomials we can state a theorem equivalent to Theorem 1.4: Theorem 3.1. Let π be a matching and p be an integer between 1 and |π | − 1 such that m p (π ) = 0, and write π = α ◦ β with |α | = p. We then have the following factorization:

ψπ ,− p ( z1 , . . . , z2n ) = g α ( z1 , . . . , z p , z pˆ , . . . , z1ˆ )ψβ ( z p +1 , . . . , z pˆ −1 ). In what follows we use the short notation z( O ) for the outer variables { z1 , . . . , z p , z pˆ , . . . , z1ˆ } and

z( I ) for the inner variables { z p +1 , . . . , z pˆ −1 }. 3.2. A contour integral formula

We will follow the same path as in Section 2.3. That is, we introduce a new quantity φa,− p ( z) defined by a multiple contour integral formula, and after we can obtain ψπ ,− p ( z) by:

ψπ ,− p ( z) :=



−1 Cπ ,a (τ )φa,− p ( z).

(12)

a

This new quantity φa,− p ( z) must be a generalization of the formula φa ( z), let jˆ = 2n − j + 1:

 

qzi − q−1 z j

φa,− p ( z) := kn

1 i , j  p

×

 



qzi − q−1 zjˆ

2 i , j  p

p 2n− p   

qzi − q−1 z j



i =1 j = p +1

×

···





qzi − q−1 z j



p
 n dw i i =1



2π i





p − w i )(qw i − q−1 w j )  qw i − q−1 zjˆ  , −1 z ) qz j − q−1 w i j j ai ( w i − z j ) j >ai (qw i − q j >i ( w j

j =1

454

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

where kn is a normalization constant:

 kn =

(q − q−1 )−n(n−1) if p = 0; 2 (q − q−1 )−( p −1) −(n− p )(n− p −1) otherwise,

and the contours of integration surround all zi but not q±2 zi . This means, that integrating is equivalent to choose all possible combinations of poles ( w k − zi )−1 for all k  n such that i  ak . Notice that the presence of the Vandermonde implies that we cannot chose the same pole twice. In the homogeneous limit zi = 1 for all i, we get:

φa,− p (1, . . . , 1) =

···

 du i i

2π

a iu i i

(1 + τ u i )− p

 (u j − u i )(1 + τ u j + u i u j ), j >i

which is precisely Φa (τ , − p ). In fact, this is the main reason for the formula presented here. Therefore, we can write

g π ( z) =



−1 Cπ ,a (τ )φa,−|π | ( z),

(13)

a

where the sum runs over all matchings a. In fact, we will use this equation as definition of g π ( z). 3.3. Some properties of φa,− p ( z) In Section 2.3, we have seen that φa ( z) are useful because they are homogeneous polynomials with a certain degree which obey to the wheel condition, thus they span Vn and we can expand ψπ ( z) as a linear combination of φa ( z). We hope that we can find some properties of φa,− p ( z) which allow us to apply similar methods. Let us start with the polynomiality: Proposition 3.2 (Polynomiality). The function φa,− p ( z1 , . . . , z2n ) is a homogeneous polynomial on the variables zi . Proof. It is obvious that φa,− p ( z) can be written as a ratio of two polynomials. Thus, to prove that this is a polynomial, it is enough to prove that there are no poles. The proof is straightforward but tedious, thus we shall not repeat on the body of this paper, see Appendix A.1 for the details. The fact that it is homogeneous is obvious from the definition, once we know that it is a polynomial. 2 Proposition 3.3 (The individual degree). The degree of the polynomials ψa,− p ( z) at a single variable zi is given by:

δi =

⎧ ⎨0

if i = 1 or i = 1ˆ ; p−1 if 1 < i  p or pˆ  i < 1ˆ ; ⎩ n − p − 1 if p < i < pˆ .

Proof. For a certain variable zi two things can happen when we perform the contour integration. Either we chose a pole ( w k − zi )−1 for some k = 1, . . . , n or not. It is enough to compute the degree in both cases and we arrive to the desired result. 2 Proposition 3.4 (The combined degree). The degree of the polynomials ψa,− p ( z) in the outer variables is ( p − 1)2 , in the inner variables is (n − p )(n − p − 1) and in all variables is ( p − 1)2 + (n − p )(n − p − 1).

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

455

Proof. The total degree in all variables is easy to compute: δ = ( p − 1)2 + (n − p )(n − p − 1). The total degree in the inner variables (respectively outer variables) is more complex. Assume that we choose α inner poles ( w i − z j )−1 for p < j < pˆ (respectively outer poles ( w i − z j )−1 for j  p or j  pˆ ). The degree is α (2n − 2p − α ) − n + p (respectively α (2p − α ) − 2p + 1). The maximum is when α = n − p (respectively α = p), and it is equal to (n − p )(n − p − 1) (resp. ( p − 1)2 ). 2 Notice that if we choose the maximum in both sets (the outer and the inner variables) we obtain

( p − 1)2 + (n − p )(n − p − 1), which is equal to δ . Thus we can write:

φa,− p ( z) =











P i z( O ) Q i z( I ) ,

(14)

i

where P i and Q i are polynomials of total degree ( p − 1)2 and (n − p )(n − p − 1) respectively. Proposition 3.5 (The wheel condition). Let zk = q2 z j = q4 zi for p < i < j < k < pˆ . Thus, φa,− p ( z) = 0.



−1 z ) contain two zeros (if z = q2 z = q4 z ), if we want to prove Proof. The term j j i k p
Therefore the inner parts Q i ( z p +1 , . . . , z pˆ −1 ) are homogeneous polynomials with total degree (n − p )(n − p − 1) and satisfy the wheel condition, so they belong to the vector space Vn− p , that is:

φa,− p ( z) =











P a;β z( O ) ψβ z( I ) ,

(15)

β:|β|=n− p

where the sum runs over all matchings β of size n − p. 3.4. Computing P a;β In order to compute these polynomials, we need a new definition: Definition 3.6. Let a = {a1 , . . . , an } be a matching of size n. Separate it into two parts: the inner part composed by all p < ai < pˆ subtracted by p, and the outer part composed by all ai  p, and the ai  pˆ subtracted by 2(n − p ). Let c be the inner part and b the outer part. Moreover if the number of elements of c is bigger than n − p by s we add s times p to the outer part. b and c are not necessarily matchings. Write a = b • c. If m p (π ) = 0, this definition coincides with the one of π = α ◦ β . For example, let a = {1, 3, 5, 6, 7, 10} and p = 4. Then, we have b = {1, 3, 4, 6} and c = {1, 2, 3}. With this new notation, the polynomials P a;β ( z1 , . . . , z p , z pˆ , . . . , z1ˆ ) are given by the following

proposition:

Proposition 3.7.

    φb•c,− p ( z) = φb,− p z( O ) C c ,β (τ )ψβ z( I ) . β

456

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

Proof. Remember that the coefficients C a,π (τ ) are defined as C a,π (τ ) = τ −d(π ) φa (qπ ) and can be constructed using the algorithm based in a recursion formula proved in Lemma A.2:

    φa qπ = [s]τ d(π )−d(πˆ ) φaˆ qπˆ , where πˆ is a matching obtained by removing a small arch ( j , j + 1) from π , aˆ is a matching obtained from a by removing one element ai of the sequence such that ai = j, decreasing all elements bigger then j by two (ai → ai − 2 if ai > j) and by one if the element is equal to j (ai → ai − 1 if ai = j), s is the number of ai such that ai = j. Therefore we can study the polynomial φb•c,− p ( z) at the special points z = { z1 , . . . , z p , qβ , z pˆ , . . . , z1ˆ }, because this is enough to characterize the polynomial. It is not difficult to see that we obtain exactly the same recursion formula, see Lemma A.4 for the technical details. Thus, it is not hard to prove that

  φb•c,− p z1 , . . . , z p , qβ , z pˆ , . . . , z1ˆ = C c,β (τ )τ d(β) φb,− p ( z1 , . . . , z p , z pˆ , . . . , z1ˆ ), which is the object of Corollary A.5. This is equivalent to the result we wanted to prove.

2

The remaining polynomial φb,− p can be expressed by means of the polynomials g α : Proposition 3.8.

     φb,− p z( O ) = C b,α (τ ) g α z( O ) . α

Proof. If b is a matching, this proposition is equivalent to the definition of g α . Thus, the hard case is when b is not a matching. In that case C b,α (τ ) is defined by the equation φb (qα ). We know that V p is spanned by φ f ( z1 , . . . , z2p ) where f is a matching of size p. So we can write φb ( z1 , . . . , z2p ) = f R b, f (τ )φ f ( z1 , . . . , z2p ), where R b, f (τ ) is a matrix to be determined. This is equivalent to

C b,α (τ ) =



R b, f (τ )C f ,α (τ ),

(16)

f

where f and α are matchings, but not necessarily b. We shall determine an algorithm to compute the matrix R b, f (τ ), but we shall only treat the case when b i  2i − 1 for all i, but ignoring the condition b i = b j if i = j. Thus, the only “anomaly” which can occur is the existence of several b i with the same value, this is, there are some p such that {bi : bi = p } > 1. Let b be such that it has one element repeated at least twice, say bk = bk+1 = j and bk−1 < j, so apart from the term (qw k − q−1 w k+1 ) the integrand is antisymmetric on w k and w k+1 . Using the fact that

 A

qw k − q−1 w k+1

+

qw k − q−1 w k+1

(qw k − q−1 z j )(qw k+1 − q−1 z j )  qw k − q−1 w k+1 +τ =0 (qw k − q−1 z j )( w k+1 − z j ) ( w k − z j )( w k+1 − z j )

we can write

φb ( z1 , . . . , z2p ) = −φb˜ ( z1 , . . . , z2p ) − τ φbˇ ( z1 , . . . , z2p )

(17)

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

457

where bˇ is obtained from b by bk → bk − 1, and b˜ is obtained from b by bk → bk − 1 and bk+1 → bk+1 − 1. If {b i such that b i < j }  j the integral vanishes. Thus, we can follow this procedure until we get either a matching or it vanishes. Now, if we look to the expression of φb,− p ( z1 , . . . , z2p ), we can try to apply the same procedure in order to get the same recursion. Two things are essential, the vanishing conditions are the same, and if bk = bk+1 = j the integrand should be antisymmetric apart from the term (qw k − q−1 w k+1 ), which is true. Having the same recursion, we can write:

φb,− p ( z1 , . . . , z2p ) =



R b, f (τ )φ f ( z1 , . . . , z2p )

f

=



=



R b, f (τ )C f ,α (τ ) g α ( z1 , . . . , z2p )

α

f

C b,α (τ ) g α ( z1 , . . . , z2p )

α

from the first to the second equations we apply the definition of g α .

2

3.5. Final details In conclusion we have that

φb•c,− p ( z) =

 α









C b,α (τ )C c ,β (τ ) g α z( O ) ψβ z( I ) .

(18)

β

A simple consequence of the algorithm that we use to compute the coefficients C a,π (τ ) is that it can be decomposed, as shown in Corollary A.3: C a,α ◦β (τ ) = C b,α (τ )C c ,β (τ ), for a = b • c. Thus,

φa,− p ( z) =

 α









C a,α ◦β (τ ) g α z( O ) ψβ z( I ) .

β

When we compare with the formula

φa,− p ( z) =



C a,π (τ )ψπ ,− p ( z),

π

we conclude that

 ψπ ,− p ( z) =

0 g α ( z( O ) )ψβ ( z( I ) )

if m p (π ) = 0, if π = α ◦ β

(19)

is a solution of the system of equations. And that solution must be unique because C a,π (τ ) is invertible. 2 4. Sum rule The main purpose of this section is to prove Theorem 1.6:

 π : |π |=n

(−1)d(π ) G π = An and

 π : |π |=n

n 

G π = (−1)

2

A nV

2

,

458

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

the first one is a simple corollary of Theorem 6.16 at [9]:

 π : |π |=n

because it is known that d(π )

(−1)





G π (−τ ) =

Ψπ (τ ),

π : |π |=n

π : |π |=n Ψπ = A n , proved in [6], and it is easy to check that G π (−1) =

Gπ .

4.1. An integral formula The quantities G π (τ ) can be expressed by: −1 G π (τ ) = C π ,a (τ )Φa





τ , −|a|

following the definition at Section 2.4, where

  Φa τ , −|a| =

···

 |a| du i i =1

2π

a iu i i

(1 + τ u i )−|a|

 (u j − u i )(1 + τ u j + u i u j ). j >i

Notice that if we change the sign of τ and at the same time the one from the variables {u i }i , the integral change like Φa (τ , −|a|) = (−1)d(τ ) Φa (τ , −|a|), which in conjunction with Eq. (6), we obtain: G π (−τ ) = (−1)d(π ) G π (τ ). Let Ln be the set of matchings (in the form of sequences) of size n defined by a ∈ Ln if and only if ai = 2i − 1 or ai = 2i − 2 for all i. Following Section 3.3 of article [8] we can conclude that:



G π (τ ) =

π : |π |=n



  Φa τ , −|a| .

(20)

a∈Ln

This results in:



G π (τ ) =

···



π : |π |=n

du i

(1 + u i )(1 + τ u i )−n 2i −1

2π iu i

i

 (u j − u i )(1 + τ u j + u i u j ), j >i

which can be compared with the formula:





Ψπ (τ ) =

···

π : |π |=n

 du i i

2π i

(1 + u i )

 (u j − u i )(1 + τ u j + u i u j ). j >i

In [9] has been proved that:

···



=

i

···

du i 2π

−1 iu 2i i

 du i i

2π i

(1 + u i )(1 − τ u i )−n

(1 + u i )

 (u j − u i )(1 − τ u j + u i u j ) j >i



(u j − u i )(1 + τ u j + u i u j ).

(21)

j >i

The idea of the proof, which we shall not repeat here, is that both sides count Totally Symmetric Self Complementary Plane Partitions (TSSCPP) with the same weights. On the one hand, it was already known that the TSSCPP can be seen as lattice paths. In this framework, we can count them using the

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

459

Lindström–Gessel–Viennot formula, moreover, in this case we have a weight τ by vertical step. This will be equivalent to the RHS. On the other hand, we can describe the TSSCPP using a different set of lattice paths, which are called Dual Paths in [9]. This will give rise to the LHS. Corollary 4.1.



(−1)d(π ) G π = An .

π : |π |=n

Proof. This is a consequence of the fact that



d(π ) Gπ . π : |π |=n Ψπ = A n and G π (−1) = (−1)

2

4.2. Alternating sign matrices Compare the RHS of Eq. (21) with the enumeration of Alternating Sign Matrices, according to the position of the 1 in the first row:

A n (x) :=

n 

A n ,i x

i −1

=

i =1

···

 n

du i

i =1

2π iu i

(1 + xu i ) 2i −1

 (u j − u i )(1 + u j + u i u j ), j >i

whose proof can be found in [11]. n   Thus, it is straightforward to see that π G π = π Ψπ (−1) = (−1) 2 A n (−1). Finally, William [17] proved that: Proposition 4.2. A n (−1) = ( A nV )2 . This completes the proof of Theorem 1.6. 5. Further questions 5.1. Solving the conjectures This paper is, in a certain way, the continuation of the article [10]. Although, we solve here two conjectures, there still two more: the fact that all coefficients of A π (t ) are positive (in fact, we have some numerical evidence that all roots of A π (t ) have negative real part) and the multiplicity of the real roots. Also, the value of G ()2m+1 it is still a conjecture. In [9] the author presents an interpretation of this value as counting a certain subset R of the Totally Symmetric Plane Partitions. In fact it is not hard to prove that this subset R is exactly the subset PnR which appears in Ishikawa’s article [14]. Therefore, this conjecture it is equivalent to the unweighted version of [14, Conjecture 4.2]. 5.2. Combinatorial reciprocity We recover here one of the ideas of [10]. The theorems proved here, together with the conjectures of [10] which remain unproved suggest that G π and A π (−t ) for t ∈ N have a combinatorial interpretation. That is, we believe that exist yet-to-be-discovered combinatorial objects indexed by the matchings π counted by |G π | or by | A π (− p )|.   Notice that, even if the sum rule of A π and G π are related ( π (−1)d(π )G π = π A π ), both quantities are essentially different, it is, they have different symmetries. On the one hand, the A π are stable in respect of the rotation and mirror symmetries. On the other hand, the G π are stable by inclusion, this is G (π ) = G π . The well-known Ehrhart polynomial i P (t ) of a lattice polytopes P , which counts the number of lattice points in t P when t is a positive integer, has an interesting property. When t is negative,

460

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

(−1)dim P i P (t ) counts the lattice points strictly inside of −t P (see [1] for instance). We believe that should be something similar in the quantities A π (t ). 5.3. A new vector space of polynomials The polynomial ψπ ( z1 , . . . , z2n ) can be seen as the solution of the quantum Knizhnik–Zamolodchikov equation, moreover they define a vector subspace of polynomials characterized by a vanish condition (the wheel condition) and a overall degree. These polynomials, are also related to the nonsymmetric Macdonald polynomials and can be constructed using some difference operators as showed by Lascoux, de Gier and Sorrell in [4]. In the same way, g π ( z1 , . . . , z2n ) span a vector subspace of polynomials with a certain fixed degree. Therefore, it will be interesting to fully characterize this vector subspace, and see if there is some other way to construct them, as the difference operator used in [4]. Acknowledgments The author is thankful to Ferenc Balogh for all the interesting discussions about this subjects and others. The author would like also to thank Philippe Nadeau, with whom he published the article that inspired this one. Appendix A. Proof of some technical lemmas A.1. Polynomiality In this appendix we prove that the quantities φa,− p ( z) are polynomials. Using Cauchy’s integral formula, the integral is not so complicated, this is, it is enough to chose poles ( w k − z j )−1 for all k such that j  ak (and no repeated j), and compute the residues on these poles. Then we sum over all possible choices. It is then clear that the result is a sum of ration of polynomials, and we want to prove that this sum has no poles. Looking to the integral formula we can identify three sources of problems:

 

(1) The term i j ai ( w i − z j )−1 , which can originate poles like ( zk − z j )−1 with k = j and j , k  ai for some i;   (2) The term i j >ai (qw i − q−1 z j )−1 , which can originate poles like (qzk − q−1 z j )−1 with k  ai < j for some i;  p (3) The term i j =1 (qz j − q−1 w i )−1 , which can originate poles like (qz j − q−1 zk )−1 with j  p and k  ai for some i. The proof follows always the same path, we isolate a certain pole, and then we find that its coefficient vanishes. A.1.1. Poles like ( zk − z j )−1 In order to obtain this pole we need to chose the residue at ( w i − z j )−1 with k  ai and/or the residue at ( w l − zk )−1 with j  al . Let us assume that we chose both of them. In the sum, there is an other term coming from the opposite choice ( w l − z j )−1 and ( w i − zk )−1 because j , k  ai , al . Notice that there is a term ( w i − w l ) which is ( z j − zk ) in the first case and ( zk − z j ), so we can write the sum of these two terms:

f ( z j , zk , w i = z j , w l = zk )( z j − zk ) + f ( z j , zk , w i = zk , w l = z j )( zk − z j )

( z j − zk )( zk − z j ) where f is some analytical function on the point zk = z j . So the pole disappears. In the case that we chose only one of the poles, the analysis is the same. We chose ( w i − z j )−1 and it is easy to verify that the pole will cancel with the one coming from choosing the pole ( w i − zk )−1 .

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

461

A.1.2. Poles like (qz j − q−1 zk ) p When k > pˆ , the term is compensated by the zero of (qw i − q−1 z ˆj ). j

 p  −1 w )−1 will be cancelled by the term −1 z ) × i j i 1i , j  p (qz i − q j =1 (qz j − q   p − 1 − 1 ( qz − q z ) ( qz − q z ) except for some terms containing z which can be 1 i i j ˆ 2 i , j  p p < j < pˆ i j



The poles from

ruled out by picking the pole ( w 1 − z1 )−1 . Finally, the only poles that can appear are coming from the term (qw i − q−1 z j )−1 with j < pˆ . If j > p, the pole correspond to w i = zk with k  ai < j, so k < j and this is ruled out by the term  −1 z ). If j  p, such term doesn’t exist. If we do not pick any pole ( w − z )−1 we j j l p


− q−1 z j ), if we pick ( w l − z j )−1 so we have that k  ai < j  al , so k < l and we have the term (qw i − q−1 w l ) = (qzk − q−1 z j ). 2 can use the term

1i , j  p (qz i

A.2. An antisymmetrization formula

s

We antisymmetrize the expression i < j (qw i − q−1 w j ). This will be useful for some Lemmas A.2 and A.4. The result is not complicated and can be obtained by several ways.  Define A as being the antisymmetrization of a function: A( f ( w 1 , . . . , w k )) = k1! σ ∈ S k sign(σ ) × f ( w σ1 , . . . , w σk ). Lemma A.1.

 A



k  

qw i − q

−1

wj



[k]!  ( w i − w j ), k! k

=

i< j

i< j

qk −q−k

where [k]! = [k][k − 1] · · · [1], and [k] := q−q−1 = qk−1 + qk−3 + · · · + q−k+1 . Proof. It is obvious that the result is a homogeneous polynomial of degree metric on the { w i }1i k it must be a multiple of k := the coefficient. Let qi , j := (qw i − q−1 w j ). On one hand we have:

 A

k  

qw i − q

−1

i< j

  wj

k 

k

2

, and as it is antisym-

i < j ( w i − w j ). The only difficulty is to find

= ck k w w k =0

= ck

k =0

 k −1 

 w i k−1 .

i =1

On the other hand, we have:

 A

k  i< j

 qi , j

= w k =0

=

1  k!

sign(σ )

σ ∈ Sk

k 1  

k!

r =1

σ ∈ Sk σr =k





qσi ,σ j w =0 k

i< j

sign(σ )

 i
qσi ,k

 j >r

qk,σ j

 i< j i , j =r



qσi ,σ j w =0 k

462

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

=

k 1  

k!

 =

r =1

k 

q

sign(σ )(−1)k−r q2r −k−1

σ ∈ Sk σr =k 2r −k−1

[k] k

wi

i =1

  k −1 1 

r

=

k −1

k!

i =1

 wi



sign( )

 ∈ S k −1



qσi ,σ j

i< j i , j =r k −1

qi , j

i< j

ck−1 k−1 .

So we have ck = [kk] ck−1 . We easily check that c 1 = 1, the result follows.

2

A.3. The recursion formulæ In this section we prove the following lemma: Lemma A.2. Let α be a matching of size n let p be a integer such that 0 < p < n, and such that m p (α ) = 0, this means that we can write α = β ◦ γ with β and γ two matchings of size, respectively, p and r := n − p. Let a be a second matching, represented by a sequence. Write a = b • c, such that b and c are the outer and inner part, respectively, and are not necessarily matchings. Clearly γ has a small arch, say ( j , j + 1). Let γˆ be the matching obtained from γ by removing the small arch ( j , j + 1). Let s count the c i such that c i = j. Let cˆ be the sequence obtained from c by keeping all c i < j, the s c i = j are transformed in (s − 1) j − 1, and all the others c i > j are transformed c i → c i − 2. We claim that

    φb•c qβ◦γ = [s]τ d(β◦γ )−d(β◦γˆ ) φb•ˆc qβ◦γˆ . The proof of this lemma is rather long but straightforward. The proof can be found in the literature, we shall reproduce it here because the understanding of the proof is important to another result. Proof of Lemma A.2. Notice that when z j +1 = q2 z j the pre-factor vanishes, thus we need to compensate it by choosing a pole ( w ζ − z j )−1 which will make appear the pole (qz j − q−1 z j +1 )−1 , so we must have j  aζ < j + 1, which means aζ = j. Assume that there are s  1 such aζ . The idea of the proof, is to pick all s different poles with this property, and we will have a lot of cancellations, turning possible an identification with a smaller integral φb•ˆc (qβ◦γˆ ). Because our formula is rather big, we will use the following shortcuts:

• We divide the z variables into two regions, and give different indices depending on the region i < j, k > j + 1; • For the variables in the integral, we divide in four regions (recall that we have s aζ = j, and let ( w ζ − z j )−1 be the chosen pole), δ is such that aδ < j,  < ζ such that a = j, η > ζ such that aη = j and θ is such that aθ > j;  • We will use the notation qik := i ,k (qzi − q−1 zk ). And equivalently, for any variable. For example  q j ,θ = θ (qz j − q−1 w θ ). Notice that we follow the rule Latin letters for the variables z and Greek letters for the variables w;

• We use also the notation 1i η , meaning that we  replace q by 1; • The symbol means smaller, for example 1ii = i >i (zi − zi ); • In order to keep simplicity, we omit every term which depends on none of the following variables { z j , z j +1 , w  , w ζ , w η }, we omit even all integration symbols except the one of w ζ ; • Define ni as the number of zi , and equivalently for all other variables; • Finally ξ = (q − q−1 ).

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

463

Let us rewrite our equation with this notation:



φb•c q

β◦γ



=

s 

ξ

−n(n−1)



dw ζ 1 δ 1ζ δ 1ηδ 1 1ζ  1η 1θ 

qi j qi j +1 q j j +1 q jk q j +1k

2π i q δ j q δ j +1 1 i 1 j q  j +1 q  k

ζ

×

1ηζ 1θ ζ 1ηη 1θ η qδ  qδζ qδ η q  q ζ qη q θ qζ η qζ θ qη η qηθ 1ζ i 1ζ j qζ j +1 qζ k 1ηi 1η j qη j +1 qηk 1θ j 1θ j +1

.

We use Lemma A.1: s   [s]!  φb•c qβ◦γ = ξ −n(n−1) qi j qi j +1 q j j +1 q jk q j +1k s!



ζ

×

dw ζ 1 δ 1ζ δ 1ηδ 1 1ζ  1η 1θ  2π i q δ j q δ j +1 1 i 1 j q  j +1 q  k

1ηζ 1θζ 1ηη 1θ η qδ  qδζ qδ η 1  1 ζ 1η q θ 1ζ η qζ θ 1η η qηθ 1ζ i 1ζ j qζ j +1 qζ k 1ηi 1η j qη j +1 qηk 1θ j 1θ j +1

.

And finally we integrate around w ζ = z j : s   [s]!  1 δ 1 j δ 1ηδ 1 1 j  1η 1θ  φb•c qβ◦γ = ξ −n(n−1) qi j qi j +1 q j j +1 q jk q j +1k s! q δ j q δ j +1 1 i 1 j q  j +1 q  k

ζ

× =

1η j 1θ j 1ηη 1θ η qδ  qδ j qδ η 1  1 j 1η q θ 1 j η q j θ 1η η qηθ 1 ji q j j +1 q jk 1ηi 1η j qη j +1 qηk 1θ j 1θ j +1

[s]! s! ×

s 

ξ −n(n−1) qi j qi j +1 q j +1k

1 δ 1 j δ 1ηδ 1 1 j  1η 1θ 

ζ

q δ j +1 1 i q  j +1 q  k

1ηη 1θ η qδ  qδ η 1  1η q θ 1 j η q j θ 1η η qηθ 1 ji 1ηi qη j +1 qηk 1θ j +1

.

Using the fact that z j = q−1 and z j +1 = q the following equalities are straightforward:

q i j +1 1 ji

1 j

= (−q)ni ; 1 jη q η j +1

q  j +1

1 jδ

= (−q)−n ; q jθ

= (−q)−nη ;

1θ j +1

q δ j +1

= (−q)−nδ ;

= (−q)−nθ .

We obtain s   1 δ 1ηδ 1 1η 1θ  [s]!  −n(n−1) φb•c qβ◦γ = (−q)ni −nδ −n −nη −nθ ξ qi j q j +1k s! 1 i qk 1ηi qηk

ζ

× 1ηη 1θ η qδ  qδη 1  1η q θ 1η η qηθ = (−q)ni −(n−1)

[s]  s

s

ξ −n(n−1) qi j q j +1k

ζ

× qδ  qδ η q  qη q θ qη η qηθ .

1 δ 1ηδ 1 1η 1θ  1ηη 1θ η 1 i qk 1ηi qηk

464

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

This is exactly the φb•ˆc (qβ◦γˆ ):

    φb•c qβ◦γ = (−q)ni −(n−1) [s]ξ −2(n−1) qi j q j +1k φb•ˆc qβ◦γˆ    2    = (−q)−(n−1) [s]ξ −2(n−1) q −1 − q 2 z i q − q−1 zk φb•ˆc qβ◦γˆ . i

k

Finally, if we replace each zi = q±1 and z j = q±1 , we get that:



q

−1





− q zi = −ξ qτ ξ 2

if zi = q−1 , if zi = q,



2

q −q

−1



zk =



−τ ξ qξ

if zk = q−1 , if zk = q.

Is a simple exercise to check that all factors q, (−1) and ξ cancel. We can also check that the number of zi = q plus the number of zk = q−1 is exactly d(β ◦ γ ) − d(β ◦ γˆ ), proving the lemma. 2 We have seen that the ψα ( z) form a basis ofthe vector space Vn . φa ( z1 , . . . , z2n ) lives at the vector space Vn too, so that we can write as φa ( z) = α C a,α (τ )ψα ( z). And this coefficients can be identified using the relation φa (qα ) = C a,α (τ )τ d(α ) . Thus, this lemma is telling us how we construct these C a,α (τ ): we pick a small arch at α , we apply the recursion ˆ , and the whole thing is multiplied by [s]τ d(α )−d(αˆ ) . Thus, the and we get smaller matchings aˆ and α coefficient C a,α (τ ) it will be the multiplication by [s] at each step, because the power on τ it will be exactly τ d(a) . Following this procedure, we can see what happen in the case that α = β ◦ γ : Corollary A.3. The coefficients C b•c ,β◦γ (τ ) obey to the simple decomposition

C b•c ,β◦γ (τ ) = C b,β (τ )C c ,γ (τ ). Proof. On one hand, if we apply the algorithm n times, we get:

  φb•c qβ◦γ = τ d(β◦γ ) C b•c,β◦γ (τ ). On the other hand, we can do the recursion starting with the r inner variables, but the algorithm is independent of all details, so:

    φb•c qβ◦γ = τ d(β◦γ )−d(β) C c,γ (τ )φb qβ = τ d(β◦γ ) C c,γ (τ )C b,β (τ ). Notice that b is not necessarily a matching, but nevertheless the coefficient C b,β (τ ) is defined by

ψb (qβ ) and similarly for c. This completes the proof.

2

Now we repeat the procedure for the case φa,− p ( z). The point is that, this polynomial is proportional to ψγ ( z( I ) ), where the γ are of size r. We then check that the algorithm of Lemma A.2 remains essentially the same, i.e. Lemma A.4. Let α and a be two matching as in Lemma A.2, thus α = β ◦ γ and a = b • c. Let ( j , j + 1) be a small arch on γ . γˆ , cˆ and s are defined as in Lemma A.2. We claim that

    φb•c,− p zh , qγ , zl = [s]τ d(γ )−d(γˆ ) φb•ˆc,− p zh , qγˆ , zl .

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

465

The proof will follow exactly the same steps as the one for Lemma A.2, though there are some details that differ. WE should repeat it here for sake of completeness. Proof of Lemma A.4. We use all definitions made on the proof of Lemma A.2, with only one difference:

• We divide the z variables in four regions: h  p, p < i < j, j + 1 < k < pˆ and l  pˆ . We rewrite the expression, using the new notation and applying Lemma A.1: s   [s]!  2 φb•c,− p zh , qγ , zl = ξ −r (r −1)−( p −1) qh j qh j +1 qi j qi j +1 q j j +1 q jk q j +1k s! ζ

dw ζ 1θ η 1θ ζ 1θ  1ηη 1ηζ 1η 1ηδ 1ζ  1ζ δ 1 1 δ ×

2π i q δ j q δ j +1 1 h 1 i 1 j q  j +1 q  k q  k 1ζ h 1ζ i 1ζ j

×

qηθ qζ θ q θ 1η η 1ζ η 1η qδ η 1 ζ qδζ 1  qδ 

ql qζ l qηl

qζ j +1 qζ k qζ l 1ηh 1ηi 1η j qη j +1 qηk qηl 1θ j 1θ j +1 qh qhζ qhη

.

We integrate around w ζ = z j : s   [s]!  2 φb•c,− p zh , qγ , zl = ξ −r (r −1)−( p −1) qh j +1 qi j qi j +1 q j +1k s!

ζ

×

1θ η 1θ  1ηη 1η 1ηδ 1 j  1 j δ 1 1 δ qδ j +1 1 h 1 i q j +1 qk qk 1 jh 1 ji

qηθ q j θ q θ 1η η 1 j η 1η qδ η 1  qδ  ql qηl × 1ηh 1ηi qη j +1 qηk qηl 1θ j +1 qh  qh η

= (−q)nh +ni (−q)−(n−1) [s]ξ −r (r −1)−( p −1) qi j q j +1k 2

×

1θ η 1θ  1ηη 1η 1ηδ 1 1 δ qηθ q θ qη η qη qδ η q  qδ  ql qηl , 1 h 1 i q  k q  k 1ηh 1ηi qηk qηl qh  qh η

but nh = p. Now we can identify φb•ˆc,− p ( zh , qγˆ , zl ):

     2    φb•c,− p zh , qγ , zl = (−q)−(r −1) [s]ξ −2(r −1) q −1 − q 2 z i q − q−1 zk φb•ˆc ,− p zh , qγˆ , zl = [s]τ

d(γ )−d(γˆ )



i

 φb•ˆc,− p zh , qγˆ , zl .

k

2

Now, we know that φa,− p ( z1 , . . . , z2n ) lives at the vector space Vr if we ignore the outer variables, thus we can write it as linear combination of the ψγ ( z p +1 , . . . , z pˆ −1 ) and the coefficients will be a function on { z1 , . . . , z p , z pˆ , . . . , z1ˆ }. Using Lemma A.4, we can construct these coefficients: Corollary A.5. Using the last lemma in the inner part we get:

  φb•c zh , qγ , zl = τ d(γ ) C c,γ (τ )φb ( zh , zl ). The proof is straightforward.

466

T. Fonseca / Advances in Applied Mathematics 51 (2013) 446–466

References [1] M. Beck, S. Robins, Computing the Continuous Discretely, Undergrad. Texts Math., Springer, New York, 2007. [2] L. Cantini, A. Sportiello, Proof of the Razumov–Stroganov conjecture, J. Combin. Theory Ser. A 118 (2011) 1549–1574. [3] F. Caselli, C. Krattenthaler, B. Lass, P. Nadeau, On the number of fully packed loop configurations with a fixed associated matching, Electron. J. Combin. 11 (2004), Research paper 16. [4] J. de Gier, A. Lascoux, M. Sorrell, Deformed Kazhdan–Lusztig elements and Macdonald polynomials, J. Combin. Theory Ser. A 119 (2012) 183–211. [5] P. Di Francesco, Totally symmetric self-complementary plane partitions and the quantum Knizhnik–Zamolodchikov equation: a conjecture, J. Stat. Mech. Theory Exp. (2006), 14 pp. [6] P. Di Francesco, P. Zinn-Justin, Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule, Electron. J. Combin. 12 (2005), Research paper 6, 27 pp. [7] P. Di Francesco, P. Zinn-Justin, Quantum Knizhnik–Zamolodchikov equation: reflecting boundary conditions and combinatorics, J. Stat. Mech. Theory Exp. (2007), 30 pp. [8] P. Di Francesco, P. Zinn-Justin, Quantum Knizhnik–Zamolodchikov equation, totally symmetric self-complementary plane partitions and alternating sign matrices, Theoret. and Math. Phys. 154 (2008) 331–348. [9] T. Fonseca, Alternating sign matrices, completely packed loops and plane partitions, Ph.D. thesis, Université Pierre et Marie Curie, 2010. [10] T. Fonseca, P. Nadeau, On some polynomials enumerating fully packed loop configurations, Adv. in Appl. Math. 47 (2011) 434–462. [11] T. Fonseca, P. Zinn-Justin, On the doubly refined enumeration of alternating sign matrices and totally symmetric selfcomplementary plane partitions, Electron. J. Combin. 15 (2008), Research paper 81. [12] T. Fonseca, P. Zinn-Justin, On some ground state components of the O(1) loop model, J. Stat. Mech. Theory Exp. (2009), 29 pp. [13] I. Frenkel, N. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992) 1–60. [14] M. Ishikawa, On refined enumerations of totally symmetric self-complementary plane partitions I, arXiv:math/0602068, 2006. [15] A. Razumov, Y. Stroganov, Combinatorial nature of the ground-state vector of the O (1) loop model, Teoret. Mat. Fiz. 138 (2004) 395–400. [16] F. Smirnov, A general formula for soliton form factors in the quantum sine-Gordon model, J. Phys. A 19 (1986) L575–L578. [17] N. Williams, An alternating sum of alternating sign matrices, Rose–Hulman Undergrad. Math. J. 9 (2008), 6 pp. [18] P. Zinn-Justin, Six-vertex, Loop and Tiling Models: Integrability and Combinatorics, LAP Lambert Academic Publishing, 2010. [19] J.-B. Zuber, On the counting of fully packed loop configurations: Some new conjectures, Electron. J. Combin. 11 (2004), Research paper 13.