Totally Symmetric Self complementary Plane Partitions

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Partition Function

In order to solve the ASM enumeration problem, it is convenient to generalize it by considering weighted enumerations. This amounts to computing the partition function of the 6-Vertex model, that is the summation over 6-V congurations with DWBC such that to each vertex is given a statistical weight, as shown in Figure 2.2, depending on n horizontal spectral parameters (one for each row) fy1; y2; : : : ; yng, n vertical spectral parameters fyn+1; yn+2; : : : ; y2ng and one global parameter q. This computation
was performed by Izergin [29], using recursion relations written by Korepin [40], and the result is an n n determinant (IK determinant). It is a symmetric function of the set fy1; : : : yng and of the set of fyn+1; : : : ; y2ng. Much later, it was observed by Stroganov [77] and Okada [58] that when q = e2i=3, the partition function is totally symmetric, i.e. symmetric in the full set fy1; : : : ; y2ng. Here, we describe this procedure using the language of Quantum Integrable Systems.

Totally Symmetric Self complementary Plane Partitions

A problem of interest is the enumeration of plane partitions that have some specic symmetries. The Totally Symmetric Self-Complementary Plane Partitions (TSSCPP) form one of these symmetry classes. In the pictorial representation, they are Plane Partitions inside a 2n2n2n cube which are invariant under the following symmetries: all permutations of coordinates of boxes; and taking the complement, that is putting cubes where they are absent and vice versa, and ipping the resulting set of cubes to form again a Plane Partition1. In Figure 3.2 we can see an example of a TSSCPP.

Non-Intersecting Lattice Paths

Plane partitions can also be represented as non-intersecting lattice paths. Before explaining how we can translate them, let us see brie y what they are. Let G be a locally nite graph, here it will always be a lattice. Let A = fA1; : : : ;Ang and B = fB1; : : : ;Bng be two collections of vertices, called initial and nal points, respectively. A familly of Non-Intersecting Lattice Paths (NILP) is a set of n paths, dened in G, going from A to B that do not touch one another.

ASM counting as an integral formula

In this section, using the partition function obtained in Section 2.2.2, we get an integral formula for the quantity A~n(x; y) and we prove that it is identical to a certain integral formula which counts NILP.
The rst step is to modify the spectral parameters zi = q􀀀1yi for 1 i n and zi = qyi for n < i 2n. In this way, the homogeneous limit corresponds to zi = 1 for all i. Moreover for q3 = 1, the partition function Zn is symmetric in the set fz1; : : : ; z2ng. The weights become a = q2zi 􀀀 q􀀀2zj.

Study of entries of the type ()p

In general, the computation of the polynomials is complicated, and there is no general closed formula. But there are some exceptions. In this section we study the polynomials indexed by congurations of the type ()p, i.e. a given conguration surrounded by p parentheses (see Figure 5.1 for an example). In the last subsection we present some properties of the polynomials for high p. In all that follows, is a link pattern of size 2r, so that ()p has size 2n with n = r+p.

Table of contents :

1 CPL and the qKZ equation 
1.1 Completely Packed Loops
1.1.1 Bond percolation
1.1.2 Connectivity
1.1.3 Transfer Matrix and Integrability
1.1.4 The Hamiltonian and the Temperley{Lieb algebra
1.1.5 A stochastic process in the link patterns
1.2 The XXZ Spin Chain Model
1.2.1 The quantum algebra Uq(su(2))
1.2.2 Translation between XXZ Spin Chain model and CPL
1.3 The quantum Knizhnik{Zamolodchikov equation
1.3.1 The operator Si
1.4 Finding a solution
1.4.1 Building a solution
1.4.2 A vector space
1.5 Contour integral formul
1.5.1 The action of the qKZ equation
1.5.2 Space
1.5.3 Link Patterns
1.6 A third basis
1.6.1 A new integral formula
1.6.2 Basis transformation
1.6.3 The homogeneous limit
1.7 An example
2 ASM and 6-Vertex model 
2.1 Alternating Sign Matrices
2.2 6-Vertex Model
2.2.1 Square Ice Model and a path model
2.2.2 Partition Function
2.2.3 The case q3 = 1
2.2.4 The homogeneous limit
2.3 Fully Packed Loops
2.3.1 The Razumov{Stroganov{Cantini{Sportiello theorem
2.3.2 Some symmetry classes
3 TSSCPP 
3.1 Plane Partitions
3.1.1 Totally Symmetric Self complementary Plane Partitions
3.2 Non-Intersecting Lattice Paths
3.2.1 The LGV formula
3.3 TSSCPP as NILP
3.4 Descending Plane Partitions
4 ASM and TSSCPP 
4.1 Generating functions
4.1.1 NILP
4.1.2 ASM
4.2 The conjecture
4.3 Integral formul
4.3.1 ASM counting as an integral formula
4.3.2 NILP counting as an integral formula
4.3.3 Equality between integral formul
4.4 The original conjecture
5 CPL and PP 
5.1 Some notation
5.2 Study of entries of the type ()p
5.2.1 a-Basis
5.2.2 Reduction to size r
5.2.3 Expansion for high p
5.2.4 Sum rule
5.2.5 A NILP formula
5.2.6 Punctured{TSSCPP
5.2.7 Limit shape
5.3 Study of entries of the type (p
5.3.1 Basis transformation
5.3.2 NILP
5.3.3 Punctured{TSSCPP
5.3.4 Limit shape
5.4 Further questions
5.4.1 Zuber’s conjectures
5.4.2 Limit shape
6 On the polynomials ()p : some conjectures 
6.1 Notes on the FPL case
6.2 The conjectures
6.2.1 Combinatorics
6.2.2 The case = 1
6.2.3 Generic
6.3 Contour integral formula for G
6.3.1 Dual paths
6.3.2 The sum of G
6.3.3 Computing G()n
6.4 The rst root
6.4.1 The proof
6.5 The subleading term of the polynomials
6.5.1 First proof
6.5.2 Second proof
6.5.3 Application to hook length products
6.6 The leading term of (; p)
6.7 Further questions
6.7.1 Solving the conjectures
6.7.2 Combinatorial reciprocity
6.7.3 Consequences of the conjectures
A A vector space 
B Proof of Lemma 2.27 
C Anti-symmetrization formul 
C.1 Proof of the identity (5.20)
C.1.1 The general case
C.1.2 Integral version
C.1.3 Homogeneous Limit
C.2 Proof of the identity (6.15)
D Examples of ()p 
E Equivalence of two denitions 
F Resume en francais 
F.1 Introduction
F.2 Boucles denses et qKZ
F.2.1 Boucles denses
F.2.2 Connectivite
F.2.3 La matrice R et l’integrabilite
F.2.4 Le cas homogene
F.2.5 La cha^ne de spins XXZ
F.2.6 L’equation de Knizhnik{Zamolodchikov quantique
F.2.7 Des integrales de contour
F.2.8 La limite homogene
F.3 Matrices a Signes Alternants et le modele a 6 vertex
F.3.1 Matrices a Signes Alternants
F.3.2 Modele a 6 vertex
F.3.3 La fonction de partition
F.3.4 Boucles compactes
F.4 Partitions planes
F.4.1 Partitions planes totalement symetriques et auto-complementaires
F.4.2 Chemins values non intersectants
F.5 ASM et TSSCPP
F.5.1 Une conjecture ranee
F.5.2 Esquisse de la preuve
F.6 Boucles denses et partitions planes
F.6.1 Polynomialite
F.6.2 La somme des ()p
F.6.3 La somme des (p
F.7 Des polyn^omes ()p : nouvelles conjectures
F.7.1 Denitions
F.7.2 Les conjectures
F.7.3 Questions ouvertes
G Resumo em portugu^es 
G.1 Introduc~ao
G.2 Lacetes densos e qKZ
G.2.1 Lacetes densos
G.2.2 Conectividade
G.2.3 A matriz R e a integrabilidade
G.2.4 Caso homogeneo
G.2.5 A cadeia de spins XXZ
G.2.6 A equac~ao Knizhnik{Zamolodchikov qu^antica
G.2.7 Integrais de contorno
G.2.8 O limite homogeneo
G.3 Matrizes de Sinal Alternante e o modelo 6-vertices
G.3.1 Matrizes de Sinal Alternante
G.3.2 Modelo 6-vertices
G.3.3 A func~ao de partic~ao
G.3.4 Lacetes compactos
G.4 Partic~oes planas
G.4.1 Partic~oes planas totalmente simetricas e auto-complementares .
G.4.2 Caminhos ponderados n~ao intersectantes
G.5 ASM e TSSCPP
G.5.1 Uma conjectura renada
G.5.2 Esquisso da prova
G.6 Lacetes densos e partic~oes planas
G.6.1 Polinomialidade
G.6.2 A soma dos ()p
G.6.3 A soma dos (p
G.7 Polinomios ()p : novas conjecturas
G.7.1 Algumas denic~oes
G.7.2 As conjecturas
G.7.3 Quest~oes abertas
References 

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