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On the Doubly Refined Enumeration ofAlternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions Tiago Fonseca∗LPTHE CNRS, UMR 7589, Univ Pierre et Marie Curie-Par

On the Doubly Refined Enumeration of

Alternating Sign Matrices and Totally Symmetric

Self-Complementary Plane Partitions

Tiago Fonseca∗LPTHE (CNRS, UMR 7589), Univ Pierre et Marie Curie-Paris6,

75252 Paris Cedex, Francefonseca @ lpthe.jussieu.fr

Paul Zinn-Justin†

LPTMS (CNRS, UMR 8626), Univ Paris-Sud,

91405 Orsay Cedex, France;

and LPTHE (CNRS, UMR 7589), Univ Pierre et Marie Curie-Paris6,

75252 Paris Cedex, Francepzinn @ lpthe.jussieu.frSubmitted: Mar 26, 2008; Accepted: Jun 5, 2008; Published: Jun 13, 2008

Abstract

We prove the equality of doubly refined enumerations of Alternating SignMatrices and of Totally Symmetric Self-Complementary Plane Partitions usingintegral formulae originating from certain solutions of the quantum Knizhnik–Zamolodchikov equation

∗ The authors thank N Kitanine for discussions, and J.-B Zuber for a careful reading of the manuscript.

† PZJ was supported by EU Marie Curie Research Training Networks “ENRAGE” CT-2004-005616, “ENIGMA” MRT-CT-2004-5652, ESF program “MISGAM” and ANR program

MRTN-“GIMP” ANR-05-BLAN-0029-01.

2.1 Alternating Sign Matrices 3

2.2 6-Vertex model 4

2.3 Totally Symmetric Self-Complementary Plane Partitions 4

2.4 Non-Intersecting Lattice Paths 6

3 The conjecture 8 3.1 ASM generating function 8

3.2 NILP generating function 9

3.3 The conjecture 10

4 The proof 10 4.1 ASM counting as the partition function of the 6-Vertex model 10

4.2 Integral formula for refined ASM counting 13

4.3 Integral formula for refined NILP counting 16

4.4 Equality of integral formulae 18

A Formulating the conjecture directly in terms of TSSCPPs 20 A.1 Extending the theorem 20

A.2 The conjecture in terms of TSSCPPs 22

B Properties of the 6-Vertex model partitionfunction 23 B.1 Korepin recursion relation 24

B.2 Cubic root of unity case 26

C The space of polynomials satisfying the wheel condition 27 D An antisymmetrization formula 29 D.1 The general case 29

D.2 Integral version 32

D.3 Homogeneous Limit 33

It is the purpose of this work to revisit an old problem using some new ideas The old problem is the interconnection between two distinct classes of combinatorial objects whose enumerative properties are intimately related: Alternating Sign Matrices and Plane Partitions [2] The new ideas come from recent developments in the so-called Razumov–Stroganov conjecture (formulated in [19]; see also [1, 3]) The Razumov– Stroganov conjecture identifies the entries of the Perron–Frobenius vector of a certain

stochastic matrix with cardinalities of subsets of Alternating Sign Matrices, the latterbeing reinterpreted as configurations of a certain two-dimensional statistical model(so-called Fully Packed Loops) Even though this statement is still a conjecture, someprogress has been made in this area in a series of papers by Di Francesco and Zinn-Justin, starting with [4] The method they used was, as it turned out, equivalent tofinding appropriate polynomial solutions of the quantum Knizhnik–Zamolodchikovequation [5] Integral representations for these and their relation to plane partitionenumeration were discussed in [6]; we shall use these integral formulae in the presentwork (noting that these can be considered as purely formal integrals, so they aresimply a way of encoding generating functions).

The paper is organized as follows In section 2, we define the various combinatorialobjects and corresponding statistical models that will be needed In section 3, weformulate the main theorem of the paper: the equality of doubly refined enumerations

of Alternating Sign Matrices and of Totally Symmetric Self-Complementary PlanePartitions Section 4 contains the proof, based on the use of integral formulae.Finally, the appendices contain various technical results that are needed in the proof.Note that even though we use some concepts and methods from exactly solvablestatistical models, this paper is self-contained and all proofs are purely combinatorial

in nature

In this section we define the various models that appear in this work There aretwo distinct models On the one hand we have Alternating Sign Matrices (ASMs)which are in bijection with configurations of the 6-Vertex model (also known as icemodel) with Domain Wall Boundary Conditions, as well as with Fully Packed Loopconfigurations (FPL) Here we only discuss ASMs and 6-V model

On the other hand we have Totally Symmetric Self-Complementary Plane tions, which are in bijection with a certain class of Non-Intersecting Lattice Paths

An Alternating Sign Matrix (ASM) is a square matrix made of 0s, 1s and -1s suchthat if one ignores 0s, 1s and -1s alternate on each row and column starting andending with 1s Here are all 3 × 3 ASMs:

Figure 1: The 6-Vertex Model is defined on a n × n grid To each link in the network

we associate an arrow which can take two directions, the only constraint being that

at each site there are two arrows pointing in and two arrows pointing out (this leaves

6 possible vertex configurations) We are only interested in the configurations suchthat the arrows at the top and at the bottom are pointing out and the arrows at theleft and the right are pointing in Here we draw all states possibles for n = 3

Thus, there are exactly 7 ASMs of size n = 3

These matrices have been studied by Mills, Robbins and Rumsey since the early1980s [14, 15, 21, 16] It was then conjectured that An, the number of ASMs of size

This was subsequently proved by Zeilberger in 1996 in an 84 page article [23]

A shorter proof was given by Kuperberg [12] in 1998 The latter is based on theequivalence to the 6-V model, which we shall also use here

Let us now turn to the 6-Vertex Model The model consists in a square grid of size

n × n in which each edge is given an orientation (an arrow), such that at each vertexthere are two arrows pointing in and two arrows pointing out We use here somevery specific boundary conditions (Domain Wall Boundary Conditions, DWBC): allarrows at the left and the right are pointing in and at the bottom and the top arepointing out

On figure1we draw all the possible configurations at n = 3 There are once again

7 configurations of size n = 3 Indeed, there is an easy bijection between ASMs and6-V configurations with DWBC, which is described schematically on figure 2

We describe here Plane Partitions in two different ways, either pictorially or as arrays

of numbers

PSfrag replacements

Figure 2: Rules to replace each vertex of a 6-V configuration with a 0 or ±1 versely, one can consistently build a 6-V configuration from an ASM starting fromthe fixed arrows on the boundary, continuing arrows through the 0s and reversingthem through the ±1

Con-Pictorially, a plane partition is a stack of unit cubes pushed into a corner (gravitypushing them to the corner) and drawn in isometric perspective, as examplified onfigure3

An equivalent way of describing these objects is to form the array of heights ofeach stack of cubes In this formulation the effect of “gravity” is that each number

in the array is less or equal than the numbers immediately above and to the left Forexample the plane partition on figure 3may be translated into the array

755317433642121111Plane partitions were first introduced by MacMahon in 1897 A problem of inter-est is the enumeration of plane partitions that have some specific symmetries TheTotally Symmetric and Self-Complementary Plane Partitions (TSSCPPs) are one ofthese symmetry classes In the pictorial representation, they are Plane Partitionsinside a 2n × 2n × 2n cube which are invariant under the following symmetries: allpermutations of the axes of the cube of size 2n×2n×2n; and taking the complement,that is putting cubes where they are absent and vice versa, and flipping the resultingset of cubes to form again a Plane Partition

Alternatively, they can be described as 2n × 2n arrays of heights In the n = 3

Figure 3: We can see a plane partition (PP) as a stack of unit cubes pushed into acorner.

case, we have, once again, 7 possible configurations:

Another important class of objects is the Non-Intersecting Lattice Paths (NILPs).These paths are defined in a lattice and connect a set of initial points to a set of finalpoints following certain rules (see Ref [13,7] for the general framework) The mostimportant feature of NILPs is that the various paths do not touch one another

0 0

0 0 0

1 1

1 0

Figure 4: Reformulation of TSSCPPs as NCLPs, in the example of size n = 3 Ifthe origin is at the upper right corner, then at each point (0, −i), i ∈ {0, , n − 1},begins a path which can only go upwards or to the right, and stops when it reachesthe diagonal (j, −j), in such a way that the numbers below/to the right of it areexactly those less or equal to n − i

In order to better understand the bijection between NILPs and TSSCPPs, it isconvenient to consider an intermediate class of objects: Non-Crossing Lattice Paths(NCLPs), which are similar to NILPs except for the fact the paths are allowed toshare a common site, although they are still forbidden to cross each other

We proceed with the description of the bijection between TSSCPPs and a class

of NCLPs Each TSSCPP is defined by a subset of numbers of the arrays of (2.2), apossible choice is the triangles at the bottom right:

• The ith path begins at (i, −i);

• The vertical steps are conserved and the horizontal steps (→) are replaced bydiagonal steps (%)

An example (n = 3) is shown on figure 5

Our last modification is the addition of one extra step to all paths To the firstpath we add a diagonal step, as for the other paths the choice is made such thatthe difference between the final point of two consecutive paths is an odd number, asexamplified on figure 6

Figure 5: We transform our NCLPs into NILPs: the starting point is now shifted tothe right, and the horizontals steps become diagonal steps.

Figure 6: To each path we add one extra step in order that two final points utive differ by an odd number The first extra step is diagonal

Various conjectures have been made to connect ASMs and TSSCPPs Building onthe already mentioned ASM conjecture by Mills and Robbins, which says that thenumber of ASMs of size n is equal to the number of TSSCPPs of size 2n (andwhich is now a theorem), there are conjectures about “refined” enumeration Beforedescribing them we need some more definitions

Each ASM, as can be easily proven, has one and only one 1 on the first row and onthe last row It is natural to classify ASMs according to their position Therefore,

we count the ASMs of size n with the first 1 in the ith position and the last 1 in the

An(x, y) :=X

i,j

An,i,jxi−1yj−1 (3.3)

Some trivial symmetries.

By reflecting the ASMs horizontally and vertically one gets:

An,i,j = An,j,iwhereas by reflecting them only horizontally one gets:

An,i,j = An,n−i+1,n−j+1

Obviously these symmetries are also valid for ˜An,i,j

First we recall the definition of the type of NILPs used in this article, of size n:

• The paths are defined on the square grid Each step connects a site to aneighbor and can be either vertical (up ↑) or diagonal (up right %)

• There are n starting points with coordinates (i, −i), i ∈ {0, 1, , n − 1} Theendpoints are at (i, 0) (so that the length of the ith path is i)

• Paths do not touch each other

It is convenient to add an extra step, as explained in section2.4, defined uniquely

by the following:

• Two consecutive paths, after the extra step, differ by an odd number

• The extra step for the first path (at (0, 0)) is diagonal

Let α be a NILP, we define u0

n(α) as the number of vertical steps in the extrastep and u1

n(α) as the number of vertical step in the last step of each path (seeappendix A.1for an extended definition)

The generating function is:

3.3 The conjecture

We now present the conjecture, formulated by Mills, Robbins and Rumsey in aslightly different language (see sectionA.2for a detailed translation), whose proof isthe main focus of the present work:

Theorem The number of ASMs of size n with the 1 of the first row in the (i + 1)st

position and the 1 of last row in the (j + 1)st position is the same as the number

of NILPs (corresponding to TSSCPPs, and with the extra step) with i vertical extrasteps and j vertical steps in the last step Equivalently,

˜

An(x, y) = Un0,1(x, y)For example, at n = 3, using the ASMs given in section 2.1 and the TSSCPPsgiven on figure 6, we compute:

˜

A3(x, y) =y2+ y + xy2+ x + xy + x2y + x2

Un0,1(x, y) =y2+ xy + x2+ xy2+ x2y + y + xThis is the doubly refined enumeration Of course, by specializing one variable, onerecovers the simple refined enumeration, i.e that the number of ASMs of size n withthe 1 of the first row in the i + 1 position is the same as the number of NILPs(corresponding to the TSSCPPs and with the extra step) with i vertical extra steps:

An(x) := ˜An(x, 1) = Un0,1(x, 1) := Un0(x)and by specializing two variables, that the number of ASMs of size n is the same asthe number of TSSCPPs of size 2n:

An = An(1) = Un0(1)

model

In order to solve the ASM enumeration problem, it is convenient to generalize it

by considering weighted enumeration This amounts to computing the partitionfunction of the 6-Vertex model, that is the summation over 6-V configurations withDWBC such that to each vertex is given a statistical weight, as shown on figure 7,

PSfrag replacements

a = q−1/2w − q1/2z b = q−1/2z − q1/2w c = (q−1− q)z1/2w1/2

Figure 7: To each site configuration corresponds a statistical weight These weightsdepend on three parameters: w (resp z) which characterizes the column (resp row),and a global parameter q which will be eventually specialized to a cubic root of unity

depending on n horizontal spectral parameters (one for each row) {z1, z2, , zn}, nvertical spectral parameters {zn+1, zn+2, , z2n} and one global parameter q Thiscomputation was performed by Izergin [8], using recursion relations written by Ko-repin [11], and the result is a n × n determinant (IK determinant) It is a symmetricfunction of the set {z1, zn} and of the set of {zn+1, , z2n} Much later, it was ob-served by Stroganov [22] and Okada [17] that when q = e2πi/3, the partition function

is totally symmetric, i.e in the full set {z1, , z2n}

More precisely, if we denote by ˜Zn the partition function, and

3−n(n−1)/2Zn(1, , 1) = An (4.2)where we recall that An is the number of ASMs of size n (as explained in 2.1).The case of interest to us is when all zi = 1 except z1 and z2n:

z1 =1 + qt

q + t

z2n =1 + qu

q + u

Using the fact that Zn(z1, , z2n) is a symmetric function of its arguments (seeappendix B), we have

where An,j,k is the number of ASMs of size n such that the only 1 in the first row is

in column j and the only 1 in the last row is in column n − k + 1

The normalization factor is equal to: (−1)n(n−1)/2(−i√3)nq1+qt

q+t

q

1+qu q+u, so wecan finally compute

4.2 Integral formula for refined ASM counting

The traditional expression for the partition function of the 6-V model is the alreadymentioned IK formula We shall not use it here We shall only need the followingfacts (true at q = e2πi/3):

• Z1 = 1

• Zn(z1, , z2n) is a polynomial of degree n − 1 in each variable

• The Zn satisfy the recursion relation for all i, j = 1, , 2n

Zn(z1, , zj = q2zi, , z2n) = Y

k6=i,j

(qzi− zk)Zn−1(z1, , ˆzi, , ˆzj, , z2n)

(4.5)

We recall how to prove them in appendix B for the sake of completeness

Furthermore, we need the following lemma

Lemma 1 A polynomial P of degree n−1 in each variable z1, , z2nwhich satisfiesthe “wheel condition”

P ( , zi = z, , zj = q2z, , zk = q4z, ) = 0 for all i < j < k

is entirely determined by its cn := (2n)!/n!/(n + 1)! values at the following izations: (q 1, , q 2n) for all possible choices of {i = ±1} such that P2n

special-i=1i = 0and Pj

i=1i ≤ 0 for all j ≤ 2n

This lemma is proved in appendix C

The strategy is now to introduce a certain integral representation of the partitionfunction of the 6-V model with DWBC, say Z0

i≤2l−1(wl− zi)Q

i≥2l−1(qwl− q−1zi) (4.6)where the integration contours surround counterclockwise the zi (but not the q−2zi),and to show that Zn and Z0

n are both polynomials of degree n − 1 in each variablewhich satisfy the “wheel condition” and coincide at the cnspecializations of lemma1

Let us first check that Zn satisfies the wheel condition This is a direct quence of Eq (4.5) in which one sets zk = q4zi It is equally straightforward tocalculate Zn at the cn points of the lemma The computation goes inductively using

conse-Eq (4.5) and it is left to the reader to check that

Zn(q 1, , q 2n) = 3(n2)

We now show that Z0

n also satisfies the hypotheses of the lemma We proceed insteps

Z0

n is a polynomial of degree n − 1 in each variable

By applying the residue formula to Eq (4.6) we obtain

We can now consider the leading term in each variable zi in the summation of

Eq (4.7), depending on whether i ∈ K or not; in both cases we find a degree n − 1

Z0

n satisfies the wheel condition

Using the formula (4.7), we can verify that Z0

n is zero at zk = q2zj = q4zi for all

k > j > i: In fact, the termQ

s<r and s / ∈K(qzs− q−1zr) implies that i and j ∈ K As aconsequence of the term Q

l<m(qzk l− q−1zk m), we must have i = km and j = kl with

l < m, but, in this case, j ≤ 2l − 1 < 2m − 1 proving that Z0

n satisfies the “wheelcondition”

If we look at formula (4.7) it is straightforward that all terms are zero except for

j = km and j + 1 ≥ 2m − 1, i.e j = km = 2m − 2 Using the fact that zj+1 = q2zj,

IY

l

dwl

2πiY

l6=m

Q

i≤2l−1 i6=j,j+1

(wl− zi)Q

i≥2l−1 i6=j,j+1

(qwl− q−1zi)

n>m

(wn− zj)(qzj − q−1wn)(wn− zj)(wn− q2zj)

Y

l<m

(zj− wl)(qwl− q−1zj)(qwl− q−1zj)(qwl− qzj)

(zj − q2zj)Q

i<j(zj− zi)Q

k>j+1(qzj − q−1zk)After multiple cancellations we get:

Zn0( , zj, zj+1 = q2zj, ) = Y

i6=j,j+1

(qzj− zi)Zn−10 (z1, , zj−1, zj+2, , z2n) (4.8)The formula actually holds for both parities of j; the proof for j odd is similar.Calculating Z0

n at the cn points

Using the formula above, we can easily calculate Z0

n at the cn points of the lemma.One can always choose two consecutive variables which are (q−1, q) and apply therecursion relation above:

i = −1 Since we have Z0

1 = 1, we obtain:

Zn0 = 3(n2)

We finally conclude, by applying lemma 1, that

Zn= Zn0Starting from our new integral formula for the partition function of the 6-Vertexmodel (4.6), we are now in a position to calculate

Y

l

dul2πi

1

u2l−2l

Q

l<m(um− ul)(1 + um+ umul)(1 + ul− x)(1 + ul(1 − y))

We will derive a contour integral formula for the generating polynomial N0

10(t0, t1, ,

tn−1) of our NILPs with a weight ti per vertical step in the ithslice (between y = 1−iand y = −i) We use the Lindstr¨om–Gessel–Viennot formula [13, 7] (see also thethird chapter of [2]):

where Pi,ris the weighted sum over all possible lattice paths from (i, −i) to (r +1, 1).Such paths counts with r − i + 1 diagonal steps and 2i − r vertical ones, hence:

Y

i=1

dui2πiu2i−1i