Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
aa r X i v : . [ m a t h - ph ] D ec SIX-VERTEX, LOOP AND TILING MODELS:INTEGRABILITY AND COMBINATORICS
PAUL ZINN-JUSTIN
Abstract.
This is a review (including some background material) of the author’s work and relatedactivity on certain exactly solvable statistical models in two dimensions, including the six-vertexmodel, loop models and lozenge tilings. Applications to enumerative combinatorics and to algebraicgeometry are described.
Contents
Introduction 31. Free fermionic methods 51.1. Definitions 51.1.1. Operators and Fock space 51.1.2. gl ( ∞ ) and d gl (1) action 71.2. Schur functions 81.2.1. Free fermionic definition 81.2.2. Wick theorem and Jacobi–Trudi identity 81.2.3. Weyl formula 91.2.4. Schur functions and lattice fermions 101.2.5. Relation to Semi-Standard Young tableaux 111.2.6. Non-Intersecting Lattice Paths and Lindstr¨om–Gessel–Viennot formula 121.2.7. Relation to Standard Young Tableaux 131.2.8. Cauchy formula 131.3. Application: Plane Partition enumeration 141.3.1. Definition 141.3.2. MacMahon formula 151.3.3. Totally Symmetric Self-Complementary Plane Partitions 161.4. Classical integrability 172. The six-vertex model 182.1. Definition 192.1.1. Configurations 192.1.2. Weights 192.2. Integrability 20 .2.1. Properties of the R -matrix 202.2.2. Commuting transfer matrices 212.3. Phase diagram 212.4. Free fermion point 222.4.1. NILP representation 222.4.2. Domino tilings 232.4.3. Free fermionic five-vertex model 232.5. Domain Wall Boundary Conditions 242.5.1. Definition 252.5.2. Korepin’s recurrence relations 252.5.3. Izergin’s formula 272.5.4. Relation to classical integrability and random matrices 272.5.5. Thermodynamic limit 292.5.6. Application: Alternating Sign Matrices 303. Loop models and Razumov–Stroganov conjecture 323.1. Definition of loop models 323.1.1. Completely Packed Loops 323.1.2. Fully Packed Loops: FPL and FPL models 323.2. Equivalence to the six-vertex model and Temperley–Lieb algebra 333.2.1. From FPL to six-vertex 333.2.2. From CPL to six-vertex 333.2.3. Link Patterns 343.2.4. Periodic Boundary Conditions and twist 353.2.5. Temperley–Lieb and Hecke algebras 363.3. Some boundary observables for loop models 383.3.1. Loop model on the cylinder 383.3.2. Markov process on link patterns 383.3.3. Properties of the steady state: some empirical observations 393.3.4. The general conjecture 404. The quantum Knizhnik–Zamolodchikov equation 414.1. Basics 414.1.1. The q KZ system 414.1.2. Normalization of the R -matrix 434.1.3. Relation to affine Hecke algebra 434.2. Construction of the solution 444.2.1. q KZ as a triangular system 444.2.2. Consistency and Jucys–Murphy elements 46 .2.3. Wheel condition 474.2.4. Recurrence relation and specializations 484.2.5. Wheel condition continued 494.3. Connection to the loop model 504.3.1. Proof of the sum rule 504.3.2. Case of few little arches 514.4. Integral formulae 544.4.1. Integral formulae in the spin basis 544.4.2. The partial change of basis 544.4.3. Sum rule and largest component 554.4.4. Refined enumeration 575. Integrability and geometry 575.1. Multidegrees 575.1.1. Definition by induction 575.1.2. Integral formula 585.2. Matrix Schubert varieties 585.2.1. Geometric description 585.2.2. Pipedreams 595.2.3. The nil-Hecke algebra 605.2.4. The Bott–Samelson construction 625.2.5. Factorial Schur functions 625.3. Orbital varieties 635.3.1. Geometric description 635.3.2. The Temperley–Lieb algebra revisited 655.3.3. The Hotta construction 665.3.4. Recurrence relations and wheel condition 665.4. Brauer loop scheme 675.4.1. Geometric description 675.4.2. The Brauer algebra 695.4.3. Geometric action of the Brauer algebra 705.4.4. The degenerate limit 71References 73 Introduction
Exactly solvable (integrable) two-dimensional lattice statistical models have played an importantrole in theoretical physics: starting with Onsager’s solution of the Ising model, they have provided on-trivial examples of critical phenomena in two dimensions and given examples of lattice real-izations of various known conformal field theories and of their perturbations. In all these physicalapplications, one is interested in the thermodynamic limit where the size of the system tends toinfinity and where details of the lattice become irrelevant. In the most basic scenario, one considersthe infra-red limit and recovers this way conformal invariance.On the other hand, combinatorics is the study of discrete structures in mathematics. Combina-torial properties on integrable models will thus be uncovered by taking a different point of viewon them which considers them on finite lattices and emphasizes their discrete properties. Thepurpose of this text is to show that the same methods and concepts of quantum integrability leadto non-trivial combinatorial results. The latter may be of intrinsic mathematical interest, in somecases proving, reproving, extending statements found in the literature. They may also lead back tophysics by taking appropriate scaling limits.Let us be more specific on the kind of applications we have in mind. First and foremost comes theconnection to enumerative combinatorics. For us the story begins in 1996, when Kuperberg showedhow to enumerate alternating sign matrices using Izergin’s formula for the six-vertex model. Theobservation that alternating sign matrices are nothing but configurations of the six-vertex model indisguise, paved the way to a fruitful interaction between two subjects which were disjoint until then:(i) the study of alternating sign matrices, which began in the early eighties after their definition byMills, Robbins, and Rumsey in relation to Dodgson condensation, and whose enumerative propertieswere studied in the following years, displaying remarkable connections with a much older class ofcombinatorial objects, namely plane partitions; and (ii) the study of the six-vertex model, one ofthe most fundamental solvable statistical models in two-dimensions, which was undertaken in thesixties and has remained at the center of the activity around quantum integrable models ever since.One of the most noteworthy recent chapters in this continuing story is the Razumov–Stroganovconjecture, in 2001, which emerged out of a collective effort by combinatorialists and physiciststo understand the connection between the aforementioned objects and another class of statisticalmodels, namely loop models. The work of the author was mostly a byproduct of various attemptsto understand (and possibly prove) this conjecture. A large part of this manuscript is dedicated toreviewing these questions.Another interesting, related application is to algebraic combinatorics, due to the appearance ofcertain families of polynomials in quantum integrable models. In the context of the Razumov–Stroganov, they were introduced by Di Francesco and Zinn-Justin in 2004, but their true meaningwas only clarified subsequently by Pasquier, creating a connection to representation theory of affineHecke algebras and to previously studied classes of polynomials such as Macdonald polynomials.These polynomials satisfy relations which are typically studied in algebraic combinatorics, e.g.involving divided difference operators. The use of specific bases of spaces of polynomials, whichis necessary for a combinatorial interpretation, connects to the theory of canonical bases and thework of Kazhdan and Lusztig.Finally, an exciting and fairly new aspect in this study of integrable models is to try to find analgebro-geometric interpretation of some of the objects and of the relations that satisfy. The mostnaive version of it would be to relate the integer numbers that appear in our models to problemsin enumerative geometry, so that they become intersection numbers for certain algebraic varieties.A more sophisticated version involves equivariant cohomology or K-theory, which typically leads topolynomials instead of integers. The connection between integrable models and certain classes ofpolynomials with geometric meaning is not entirely new, and the work that will be described herebears some resemblance, as will be reminded here, to that of Fomin and Kirillov on Schubert andGrothendieck polynomials. However there are also novelties, including the use of the multidegree echnology of Knutson et al, and we apply these ideas to a broad class of models, resulting in newformulas for known algebraic varieties such as orbital varieties and the commuting variety, as wellas in the discovery of new geometric objects, such as the Brauer loop scheme.The presentation that follows, though based on the articles of the author, is meant to be essen-tially self-contained. It is aimed at researchers and graduate students in mathematical physics orin combinatorics with an interest in exactly solvable statistical models. For simplicity, the inte-grable models that are defined are based on the underlying affine quantum group U q ( d sl (2)), withthe notable exception of the discussion of the Brauer loop model in the last section. Furthermore,only the spin 1 / Free fermionic methods
As mentioned above, we want to spend some time defining a typical free fermionic model and toapply it to rederive some useful formulae for Schur functions, which will be needed later. We shallalso need some formulae concerning the enumeration of plane partitions, which will appear at theend of this section.1.1.
Definitions.
Operators and Fock space.
Consider a fermionic operator ψ ( z ):(1.1) ψ ( z ) = X k ∈ Z + ψ − k z k − , ψ ⋆ ( z ) = X k ∈ Z + ψ ⋆k z k − with anti-commutation relations(1.2) [ ψ ⋆r , ψ s ] + = δ rs [ ψ r , ψ s ] + = [ ψ ⋆r , ψ ⋆s ] + = 0 ψ ( z ) and ψ ⋆ ( z ) should be thought of as generating series for the ψ k and ψ ⋆k , so that z is justa formal variable. What we have here is a complex (charged) fermion, with particles, and anti-particles which can be identified with holes in the Dirac sea. These fermions are one-dimensional,in the sense that their states are indexed by (half-odd-)integers; ψ ⋆k creates a particle (or destroysa hole) at location k , whereas ψ k destroys a particle (creates a hole) at location k . e shall explicitly build the Fock space F and the representation of the fermionic operators now.Start from a vacuum | i which satisfies(1.3) ψ k | i = 0 k > , ψ ⋆k | i = 0 k < | i = · · · t t t t t t t t t td d d d d · · · Then any state can be built by action of the ψ k and ψ ⋆k from | i . In particular one can definemore general vacua at level ℓ ∈ Z :(1.4) | ℓ i = ( ψ ⋆ℓ − ψ ⋆ℓ − · · · ψ ⋆ | i ℓ > ψ ℓ + ψ ℓ + · · · ψ − | i ℓ < · · · t t t t t t t t t td d d d d ℓ · · · which will be useful in what follows. They satisfy(1.5) ψ k | ℓ i = 0 k > ℓ, ψ ⋆k | ℓ i = 0 k < ℓ More generally, define a partition to be a weakly decreasing finite sequence of non-negative integers: λ ≥ λ ≥ · · · ≥ λ n ≥
0. We usually represent partitions as
Young diagrams (also called Ferrersdiagrams): for example λ = (5 , , ,
1) is depicted as λ =To each partition λ = ( λ , . . . , λ n ) we associate the following state in F ℓ :(1.6) | λ ; ℓ i = ψ ⋆ℓ + λ − ψ ⋆ℓ + λ − · · · ψ ⋆ℓ + λ n − n + | ℓ − n i Note the important property that if one “pads” a partition with extra zeroes, then the correspondingstate remains unchanged. In particular for the empty diagram ∅ , | ∅ ; ℓ i = | ℓ i . For ℓ = 0 we justwrite | λ ; 0 i = | λ i .This definition has the following nice graphical interpretation: the state | λ ; ℓ i can be described bynumbering the edges of the boundary of the Young diagram, in such a way that the main diagonalpasses between ℓ − and ℓ + ; then the occupied (resp. empty) sites correspond to vertical (resp.horizontal) edges. With the example above and ℓ = 0, we find (only the occupied sites are numberedfor clarity) . . . − − − − ... tt tt t t tttttttt dddddddd The | λ ; ℓ i , where λ runs over all possible partitions (two partitions being identified if they areobtained from each other by adding or removing zero parts), form an orthonormal basis of asubspace of F which we denote by F ℓ . ψ k and ψ ⋆k are Hermitean conjugate of each other. ote that (1.6) fixes our sign convention of the states. In particular, this implies that when oneacts with ψ k (resp. ψ ⋆k ) on a state | λ i with a particle (resp. a hole) at k , one produces a new state | λ ′ i with the particle removed (resp. added) at k times − k .The states λ can also be produced from the vacuum by acting with ψ to create holes; payingattention to the sign issue, we find(1.7) | λ ; ℓ i = ( − | λ | ψ ℓ − λ ′ + · · · ψ ℓ − λ ′ m + m − | ℓ + m i where the λ ′ i are the lengths of the columns of λ , | λ | is the number of boxes of λ and m = λ .This formula is formally identical to (1.6) if we renumber the states from right to left, exchange ψ and ψ ⋆ , and replace λ with its transpose diagram λ ′ (this property is graphically clear). So theparticle–hole duality translates into transposition of Young diagrams.Finally, introduce the normal ordering with respect to the vacuum | i :(1.8) : ψ ⋆j ψ k : = − : ψ k ψ ⋆j : = ( ψ ⋆j ψ k j > − ψ k ψ ⋆j j < gl ( ∞ ) and [ gl (1) action. The operators ψ ⋆ ( z ) ψ ( w ) give rise to the Schwinger representationof gl ( ∞ ) on F , whose usual basis is the : ψ ⋆r ψ s : , r, s ∈ Z + , and the identity. In the first quantizedpicture this representation is simply the natural action of gl ( ∞ ) on the one-particle Hilbert space C Z + and exterior products thereof. The electric charge J = P r : ψ ⋆r ψ r : is a conserved numberand classifies the irreducible representations of gl ( ∞ ) inside F , which are all isomorphic. Thehighest weight vectors are precisely our vacua | ℓ i , ℓ ∈ Z , so that F = ⊕ ℓ ∈ Z F ℓ with F ℓ the subspacein which J = ℓ .The gl (1) current(1.9) j ( z ) = : ψ ⋆ ( z ) ψ ( z ) : = X n ∈ Z J n z − n − with J n = P r : ψ ⋆r − n ψ r : forms a [ gl (1) (Heisenberg) sub-algebra of gl ( ∞ ):(1.10) [ J m , J n ] = mδ m, − n Note that positive modes commute among themselves. This allows to define the general “Hamil-tonian”(1.11) H [ t ] = ∞ X q =1 t q J q where t = ( t , . . . , t q , . . . ) is a set of parameters (“times”).The J q , q >
0, displace one of the fermions q steps to the left. This is expressed by the formulaedescribing the time evolution of the fermionic fields: e H [ t ] ψ ( z ) e − H [ t ] = e − P ∞ q =1 t q z q ψ ( z ) e H [ t ] ψ ⋆ ( z ) e − H [ t ] = e + P ∞ q =1 t q z q ψ ⋆ ( z )(1.12)(proof: compute [ J q , ψ [ ⋆ ] ( z )] = ± z q ψ [ ⋆ ] ( z ) and exponentiate). Of course, similarly, J − q , q > q steps to the right. .2. Schur functions.
Free fermionic definition.
It is known that the map | Φ i 7→ h ℓ | e H [ t ] | Φ i is an isomorphismfrom F ℓ to the space of polynomials in an infinite number of variables t , . . . , t q , . . . . Thus, weobtain a basis of the latter as follows: for a given Young diagram λ , define the Schur function s λ [ t ]by(1.13) s λ [ t ] = h ℓ | eH [ t ] | λ ; ℓ i (by translational invariance it is in fact independent of ℓ ). In the language of Schur functions, the t q are (up to a conventional factor 1 /q ) the power sums , see section 1.2.3 below.We provide here various expressions of s λ [ t ] using the free fermionic formalism. In fact, many ofthe methods used are equally applicable to the following more general quantity:(1.14) s λ/µ [ t ] = h µ ; ℓ | eH [ t ] | λ ; ℓ i where λ and µ are two partitions. It is easy to see that in order for s λ/µ [ t ] to be non-zero, µ ⊂ λ as Young diagrams; in this case s λ/µ is known as the skew Schur function associated to the skewYoung diagram λ/µ . The latter is depicted as the complement of µ inside λ . This is appropriatebecause skew Schur functions factorize in terms of the connected components of the skew Youngdiagram λ/µ . Examples: s = t , s = t − t , s = t + t , s = t − t . s = s = t , s = t + t t + t − t t − t .1.2.2. Wick theorem and Jacobi–Trudi identity.
First, we apply the Wick theorem. Consider asthe definition of the time evolution of fermionic fields: ψ k [ t ] = e H [ t ] ψ k e − H [ t ] ψ ⋆k [ t ] = e H [ t ] ψ ⋆k e − H [ t ] (1.15)In fact, (1.12) gives us the “solution” of the equations of motion in terms of the generating series ψ ( z ), ψ ⋆ ( z ).Noting that the Hamiltonian is quadratic in the fields, we now state the Wick theorem:(1.16) h ℓ | ψ i [0] · · · ψ i n [0] ψ ⋆j [ t ] . . . ψ ⋆j n [ t ] | ℓ i = det ≤ p,q ≤ n h ℓ | ψ i p [0] ψ ⋆j q [ t ] | ℓ i Next, start from the expression (1.14) of s λ/µ [ t ]: padding with zeroes λ or µ so that they havethe same number of parts n , we can write s λ/µ [ t ] = h− n | ψ µ n − n + · · · ψ µ − eH [ t ] ψ ⋆λ − · · · ψ ⋆λ n − n + |− n i and apply the Wick theorem to find: s λ/µ [ t ] = det ≤ p,q ≤ n h− n | ψ µ p − p + eH [ t ] ψ ⋆λ q − q + |− n i It is easy to see that h− n | ψ i e H [ t ] ψ ⋆j |− n i does not depend on n and thus only depends on j − i . Letus denote it(1.17) h k [ t ] = h | eH [ t ] ψ ⋆k + | i X k ≥ h k [ t ] z k = h | eH [ t ] ψ ⋆ ( z ) | i = e P q ≥ t q z q ( k = j − i ; note that h k [ t ] = 0 for k < he final formula we obtain is(1.18) s λ/µ [ t ] = det ≤ p,q ≤ n (cid:0) h λ q − µ p − q + p [ t ] (cid:1) or, for regular Schur functions,(1.19) s λ [ t ] = det ≤ p,q ≤ n (cid:0) h λ q − q + p [ t ] (cid:1) This is known as the Jacobi–Trudi identity.By using “particle–hole duality”, we can find a dual form of this identity. We describe our statesin terms of hole positions, parametrized by the lengths of the columns λ ′ p and µ ′ q , according to(1.7): s λ/µ [ t ] = ( − | λ | + | µ | h m | ψ ⋆ − µ ′ m + m − · · · ψ ⋆ − µ ′ + eH [ t ] ψ − λ ′ + · · · ψ − λ ′ m + m − | m i Again the Wick theorem applies and expresses s λ/µ in terms of the two point-function h m | ψ ⋆i e H [ t ] ψ j | m i ,which only depends on i − j = k and is given by(1.20) e k [ t ] = ( − k h− | eH [ t ] ψ − k + | i X k ≥ e k [ t ] z k = h− | eH [ t ] ψ ( − z ) | i = e P q ≥ ( − q − t q z q The finally formula takes the form(1.21) s λ/µ [ t ] = det ≤ p,q ≤ n (cid:16) e λ ′ q − µ ′ p − q + p [ t ] (cid:17) or, for regular Schur functions,(1.22) s λ [ t ] = det ≤ p,q ≤ n (cid:16) e λ ′ q − q + p [ t ] (cid:17) This is the dual Jacobi–Trudi identity, also known as Von N¨agelsbach–Kostka identity.1.2.3.
Weyl formula.
In the following sections 1.2.3–1.2.6, we shall fix an integer n and considerthe following change of variable (this is essentially the Miwa transformation [79]) t q = q P ni =1 x qi .The Schur function becomes a symmetric polynomial of these variables x i , which we denote by s λ ( x , . . . , x n ), and we now derive a different (first quantized) formula for it.Due to obvious translational invariance of all the operators involved, we may as well set ℓ = n .Use the definition (1.6) of | λ i and the commutation relations (1.12) to rewrite the left hand side as h n | eH [ t ] | λ ; n i = e P q ≥ t q P ni =1 z qi h n | ψ ⋆ ( z ) ψ ⋆ ( z ) · · · ψ ⋆ ( z n ) | i (cid:12)(cid:12) z n + λ − z n + λ − ...z λnn where (cid:12)(cid:12) ... means picking one term in a generating series.We can easily evaluate the remaining bra-ket to be: (we now use the ℓ = 0 notation for the l.h.s.) h | eH [ t ] | λ i = e P q ≥ t q P ni =1 z qi Y ≤ i Note that the change of variables t q = q P nj =1 x qj allowsus to write eH [ t ] = n Y i =1 eφ + ( x i ) φ + ( x ) = X q ≥ x q q J q So we can think of the “time evolution” as a series of discrete steps represented by commutingoperators exp φ + ( x i ). In the language of statistical mechanics, these are transfer matrices (and theexistence of a one-parameter family of commuting transfer matrices exp φ + ( x ) is of course relatedto the integrability of the model). We now show that they have a very simple meaning in terms oflattice fermions.Consider a two-dimensional square lattice, one direction being our space Z + and one directionbeing time. In what follows we shall reverse the arrow of time (that is, we shall consider thattime flows upwards on the pictures), which makes the discussion slightly easier since products ofoperators are read from left to right. The rule to go from one step to the next according to theevolution operator exp φ + ( x ) can be formulated either in terms of particles or in terms of holes: • Each particle can go straight or hop to the right as long as it does not reach the (original)location of the next particle. Each step to the right is given a weight of x . • Each hole can only go straight or one step to the left as long as it does not bump into itsneighbor. Each step to the left is given a weight of x .Obviously the second description is simpler. An example of a possible evolution of the system withgiven initial and final states is shown on Fig. 1(a).The proof of these rules consists in computing explicitly h µ | e φ + ( x ) | λ i by applying say (1.21) for t q = q x q , and noting that in this case, according to (1.20), e n [ t ] = 0 for n > 1. This stronglyconstrains the possible transitions and produces the description above. h x xh xh x xh xxh x xh x xh xh xxh x xh x xh x xhxh x x xh xh x xhx xh x x xh xh xh Figure 1. A lattice fermion configuration and the corresponding (skew) SSYT.1.2.5. Relation to Semi-Standard Young tableaux. A semi-standard Young tableau (SSYT) of shape λ is a filling of the Young diagram of λ with elements of some ordered alphabet, in such a way thatrows are weakly increasing and columns are strictly increasing.We shall use here the alphabet { , , . . . , n } . For example with λ = (5 , , , 1) one possible SSYTwith n ≥ ∅ , , , , , = λ So a Young tableau is nothing but a statistical configuration of our lattice fermions, where theinitial state is the vacuum. Similarly, a skew SSYT is a filling of a skew Young diagram with thesame rules; it corresponds to a statistical configuration of lattice fermions with arbitrary initial andfinal states. The correspondence is exemplified on Fig. 1(b).Each extra box corresponds to a step to the right for particles or to the left for holes. The initialand final states are ∅ and λ , which is the case for Schur functions, cf (1.13). We conclude that thefollowing formula holds:(1.24) s λ ( x , . . . , x n ) = X T ∈ SSYT( λ,n ) Y b box of T x T b This is often taken as a definition of Schur functions. It is explicitly stable with respect to n inthe sense that s λ ( x , . . . , x n , , . . . , 0) = s λ ( x , . . . , x n ). It is however not obvious from it that s λ is symmetric by permutation of its variables. This fact is a manifestation of the underlying freefermionic (“integrable”) behavior. Of course an identical formula holds for the more general caseof skew Schur functions. ∧ ∧ ∧ ∧ ∧ ∧∧ ∧ ∧ ∧ ∧ ∧ ∧∧ ∧ ∧ ∧ ∧ ∧ ∧ > > > > > >> > > > > >> > > > > >> > > > > > x x x x x x xx x x x x x x ∧ ∧ ∧ ∧ ∧ ∧ ∧∧ ∧ ∧ ∧ ∧ ∧ ∧∧ ∧ ∧ ∧ ∧ ∧ ∧∧ ∧ ∧ ∧ ∧ ∧ ∧ x x x x x x xx x x x x x xh h h h h h hh h h h h h h Figure 2. Underlying directed graphs for particles and holes.1.2.6. Non-Intersecting Lattice Paths and Lindstr¨om–Gessel–Viennot formula. The rules of evolu-tion given in section 1.2.4 strongly suggest the following explicit description of the lattice fermionconfigurations. Consider the directed graphs of Fig. 2 (the graphs are in principle infinite to the leftand right, but any given bra-ket evaluation only involves a finite number of particles and holes andtherefore the graphs can be truncated to a finite part). Consider Non-Intersecting Lattice Paths (NILPs) on these graphs: they are paths with given starting points (at the bottom) and givenending points (at the top), which follow the edges of the graph respecting the orientation of thearrows, and which are not allowed to touch at any vertices. One can check that the trajectories ofholes and particles following the rules described in section 1.2.4 are exactly the most general NILPson these graphs.In this context, the Jacobi–Trudi identity (1.19) becomes a consequence of the so-called Lind-str¨om–Gessel–Viennot formula [72, 36]. This formula expresses N ( i , . . . , i n ; j , . . . , j n ), the weightedsum of NILPs on a general directed acyclic graph from starting locations i , . . . , i n to ending loca-tions j , . . . , j n , where the weight of a path is the products of weights of the edges, as(1.25) N ( i , . . . , i n ; j , . . . , j n ) = det p,q N ( i p ; j q )More precisely, in Lindstr¨om’s formula, sets of NILPs such that the path starting from i k ends at j w ( k ) get an extra sign which is that of the permutation w . This is nothing but the Wick theoremonce again (but with fermions living on a general graph), and from this point of view is a simpleexercise in Grassmannian Gaussian integrals. In the special case of a planar graph with appropriatestarting points (no paths are possible between them) and ending points, only one permutation, saythe identity up to relabelling, contributes.In order to use this formula, one only needs to compute N ( i ; j ), the weighted sum of paths from i to j . Let us do so in our problem.In the case of particles (left graph), numbering the initial and final points from left to right, wefind that the weighted sum of paths from i to j , where a weight x i is given to each right moveat time-step i , only depends on j − i ; if we denote it by h j − i ( x , . . . , x n ), we have the obviousgenerating series formula X k ≥ h k ( x , . . . , x n ) z k = n Y i =1 − zx i Note that this formula coincides with the alternate definition (1.17) of h k [ t ] if we set as usual t q = q P ni =1 x qi . Thus, applying the LGV formula (1.25) and choosing the correct initial and finalpoints for Schur functions or skew Schur functions, we recover immediately (1.18,1.19). n the case of holes (right graph), numbering the initial and final points from right to left, wefind once again that the weighted sum of paths from i to j , where a weight x i is given to each leftmove at time-step i , only depends on j − i ; if we denote it by e j − i ( x , . . . , x n ), we have the equallyobvious generating series formula X k ≥ e k ( x , . . . , x n ) z k = n Y i =1 (1 + zx i )which coincides with (1.20), thus allowing us to recover (1.21,1.22).1.2.7. Relation to Standard Young Tableaux. A Standard Young Tableau (SYT) of shape λ is afilling of the Young diagram of λ with elements of some ordered alphabet, in such a way thatboth rows and columns are strictly increasing. There is no loss of generality in assuming that thealphabet is { , . . . , n } , where n = | λ | is the number of boxes of λ . For example,1 2 6 8 93 457is a SYT of shape (5 , , , λ is the dimension of λ as an irreducible representation ofthe symmetric group, which is up to a factor n ! the evaluation of the Schur function s λ at t q = δ q .Indeed, in this case one has H [ t ] = J , and there is only one term contributing to the bra-ket h λ | e H [ t ] | i in the expansion of the exponential: s λ [ δ · ] = 1 n ! h λ | J n | i In terms of lattice fermions, J has a direct interpretation as the transfer matrix for one particlehopping one step to the left. As the notion of SYT is invariant by transposition, particles and holesplay a symmetric role so that the evolution can be summarized by either of the two rules: • Exactly one particle moves one step to the right in such a way that it does not bump intoits neighbor; all the other particles go straight. • Exactly one hole moves one step to the left in such a way that it does not bump into itsneighbor; all the other holes go straight.An example of such a configuration is given on Fig. 3.1.2.8. Cauchy formula. As an additional remark, consider the commutation of e H [ t ] and e H ⋆ [ u ] ,where H ⋆ [ u ], the transpose of H [ u ], is obtained from it by replacing J q with J − q . Using theBaker–Campbell–Hausdorff formula and the commutation relations (1.10) we find eH [ t ] eH ⋆ [ u ] = e P q ≥ qt q u q eH ⋆ [ u ] eH [ t ]or equivalently e φ + ( x ) e φ − ( y ) = − xy e φ − ( y ) e φ + ( x ) with φ ± ( x ) = P q ≥ x q q J ± q .If we now use the fact that the | λ i form a basis of F , we obtain the Cauchy formula:(1.26) h | eH [ t ] eH ⋆ [ u ] | i = X λ s λ [ t ] s λ [ u ] = Y i,j (1 − x i y j ) − = e P q ≥ qt q u q (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) x xh xh x x xhx xh x xh x xhx xh x x xh xhx x xh x xh xhx x x xh xh xh ∅ Figure 3. A lattice fermion configuration and the corresponding SYT.(a) (b) Figure 4. (a) A plane partition of size 2 × × 4. (b) The corresponding dimer configuration.with t q = q P ni =1 x qi , u q = q P ni =1 y qi .1.3. Application: Plane Partition enumeration. Plane partitions are a well-known class ofcombinatorial objects. The name originates from the way they were first introduced [74] as two-dimensional generalizations of partitions; here we shall directly define plane partitions graphically.Their study has a long history in mathematics, with a renewal of interest in the eighties [97] incombinatorics, and more recently in mathematical physics [86].1.3.1. Definition. Intuitively, plane partitions are pilings of boxes (cubes) in the corner of a room,subject to the constraints of gravity. An example is given on Fig. 4(a). Typically, we ask for thecubes to be contained inside a bigger box (parallelepiped) of given sizes.Alternatively, one can project the picture onto a two-dimensional plane (which is inevitably whatwe do when we draw the picture on paper) and the result is a tiling of a region of the plane bylozenges (rhombi with 60/120 degrees angles) of three possible orientations, as shown on the rightof the figure. If the cubes are inside a parallelepiped of size a × b × c , then, possibly drawing thewalls of the room as extra tiles, we obtain a lozenge tiling of a hexagon with sides a, b, c , which isthe situation we consider now.Note that each lozenge is the union of two adjacent triangles which live on an underlying fixedtriangular lattice. So this is a statistical model on a regular lattice. In fact, we can identify it with amodel of dimers living on the dual lattice, that is the honeycomb lattice. Each lozenge correspondsto an occupied edge, see Fig. 4(b). Dimer models have a long history of their own (most notably, asteleyn’s formula [51] is the standard route to their exact solution, which we do not use here),which we cannot possibly review here.1.3.2. MacMahon formula. In order to display the free fermionic nature of plane partitions, weshall consider the following operation. In the 3D view, consider slices of the piling of boxes byhyperplanes parallel to two of the three axis and such that they are located half-way betweensuccessive rows of cubes. In the 2D view, this corresponds to selecting two orientations among thethree orientations of the lozenges and building paths out of these. Fig. 5 shows on the left theresult of such an operation: a set of lines going from one side to the opposite side of the hexagon.They are by definition non-intersecting and can only move in two directions. Inversely, any set ofsuch NILPs produces a plane partition.At this stage one can apply the LGV formula. But there is no need since this is actually the casealready considered in section 1.3.4. Compare Figs. 5 and 1: the trajectories of holes are exactlyour paths (the trajectories of particles form another set of NILPs corresponding to another choiceof two orientations of lozenges). If we attach a weight of x i to each blue lozenge at step i , we findthat the weighted enumeration of plane partitions in a a × b × c box is given by: N a,b,c ( x , . . . , x a + b ) = h | eH [ t ] | b × c i = s b × c ( x , . . . , x a + b )where b × c is the rectangular Young diagram with height b and width c . In particular the unweightedenumeration is the dimension of the Young diagram b × c as a GL ( a + b ) representation:(1.27) N a,b,c = a Y i =1 b Y j =1 c Y k =1 i + j + k − i + j + k − q to each cube in the 3D picture. It can be shownthat this is achieved by setting x i = q a + b − i (up to a global power of q ). This way we find the q -deformed formula N a,b,c ( q ) = a Y i =1 b Y j =1 c Y k =1 − q i + j + k − − q i + j + k − Many more formulae can be obtained in this formalism. The reader may for example prove that N a,b,c = X λ : λ ≤ c s λ (1 , . . . , | {z } a ) s λ (1 , . . . , | {z } b )or that N a,b,c = det(1 + T c × b T b × a T a × c )(where T y × x is the matrix with y rows and x columns and entries (cid:0) ij (cid:1) , i = 0 , . . . , y − j =0 , . . . , x − a, b, c → ∞ , and by comparing the power of thefactors 1 − q a in the numerator and the denominator, one finds another classical formula N ∞ , ∞ , ∞ ( q ) = ∞ Y n =1 (1 − q n ) − n Note that our description in terms of paths clearly breaks the threefold symmetry of the originalhexagon. It strongly suggests that one should be able to introduce three series of parameters toprovide an even more refined counting of plane partitions. With two sets of parameters, this isin fact known in the combinatorial literature and is related to so-called double Schur functions ba Figure 5. NILPs corresponding to a plane partition.(these will reappear in section 5.2.5). The full three-parameter generalization is less well-knownand appears in [107], as will be recalled in section 4.3.2. Remark : as the name suggests, plane partitions are higher dimensional versions of partitions,that is of Young diagrams. After all, each slice we have used to define our NILPs is also a Youngdiagram itself. However these Young diagrams should not be confused with the ones obtained fromthe NILPs by the correspondence of section 1.2.1.3.3. Totally Symmetric Self-Complementary Plane Partitions. In the mathematical literature,many more complicated enumeration problems are addressed, see [97]. In particular, considerlozenge tilings of a hexagon of shape 2 a × a × a . One notes that there is a group of transformationsacting naturally on the set of configurations. We consider here the dihedral group of order 12which is consists of rotations of π/ τ is attached to every blue lozenge in the fundamental domain [28].Let us call r j the location of the endpoint of the j th path, numbered from top to bottom startingat zero. We first apply the LGV formula to write the number of NILPs with given endpoints to bedet( N i,r j ) ≤ i,j ≤ n − where N i,r = τ i − r − (cid:0) i i − r − (cid:1) = (1 + τ u ) i | u i − r − . Next we sum over them and igure 6. A TSSCPP and the associated NILP.obtain N n ( τ ) = X ≤ r The free fermionic Fock space is also important for the constructionof solutions of classically integrable hierarchies. We cannot possibly describe these important ideashere, and refer the reader to [44] and references therein for details. Since an explicit example willappear in section 2, let us simply say a few general words. Recall the isomorphism Φ 7→ h ℓ | e H [ t ] | Φ i from F ℓ to the space of polynomials in the variables t q (or equivalently to the space of symmetric igure 7. All TSSCPPs of size 1, 2, 3. Figure 8. A configuration of the six-vertex model.functions if the t q are interpreted as power sums). The resulting function will be a tau-function ofthe Kadomtsev–Petiashvili (KP) hierarchy (as a function of the t q ) for appropriately chosen | Φ i .By appropriately chosen we mean the following.In the first quantized picture, the essential property of free fermions is the possibility to writetheir wave function as a Slater determinant; this amounts to considering states which are exteriorproducts of one-particle states. Geometrically this is interpreted as saying that the state (definedup to multiplication by a scalar) really lives in a subspace of the full Hilbert space called a Grass-mannian. The equations defining this space (Pl¨ucker relations) are quadratic; these equations aredifferential equations satisfied by h ℓ | e H [ t ] | Φ i . They are Hirota’s form of the equations defining theKP hierarchy.In section 2 we shall find ourselves in a slightly more elaborate setting, which results in the Todalattice hierarchy. 2. The six-vertex model The six vertex model is an important model of classical statistical mechanics in two dimensions,being the prototypical (vertex) integrable model. The ice model (infinite temperature limit ofthe six-vertex model) was solved by Lieb [69] in 1967 by means of Bethe Ansatz, followed byseveral generalizations [68, 70, 71]. The solution of the most general six vertex model was given bySutherland [99] in 1967. The bulk free energy was calculated in these papers for periodic boundaryconditions (PBC). Here our main interest will be in a different kind of boundary conditions, theso-called Domain Wall Boundary Conditions. But first we provide a brief review of the six-vertexmodel. a b b c c Figure 9. Weights of the six-vertex model.2.1. Definition. Configurations. The six-vertex model is defined on a (subset of the) square lattice by puttingarrows (two possible directions) on each edge of the lattice, with the additional rule that at eachvertex, there are as many incoming arrows as outgoing ones. See Fig. 8 for an example, and fortwo alternative descriptions: the “square ice” version in which arrows represent which oxygen atom(sitting at each lattice vertex) the hydrogen ions (living on the edges) are closer to, with the “icerule” that exactly two hydrogen ions are close to each oxygen atom; and the “path” version inwhich one considers edges with right or up arrows as occupied, so that they form north-east goingpaths. Around a given vertex, there are only 6 configurations of edges which respect the arrowconservation rule, see Fig. 9, hence the name of the model.2.1.2. Weights. The weights are assigned to the six vertices, see Fig. 9. Thus the partition functionis given by Z = X configurations Y vertex (weight of the vertex)An additional remark is useful. With any fixed boundary conditions, one can show that thedifference between the numbers of vertices of the two types c is constant (independent of theconfiguration). This means that only the product c = c c of their two weights matters.Let us denote similarly a = a a and b = b b . One can write a = ae + E x + E y a = ae − E x − E y b = be − E x + E y b = be + E x − E y and consider that a , b , c are the weights of the vertices, while E x , E y are electric fields. In whatfollows, we shall consider by default the model without any electric field, where the Boltzmannweights are invariant by reversal of every arrow and a = a = a , a = a = a , a = a = a ; andsometimes comment on the generalization to non-zero fields.There is another way to formulate the partition function, using a transfer matrix. In order to setup a transfer matrix formalism, we first need to specify the boundary conditions. Let us considerdoubly periodic boundary conditions in the two directions of the lattice, so that the model is definedon lattice of size M × L with the topology of a torus. Then one can write Z = tr T ML where T L is the 2 L × L transfer matrix which corresponds to a periodic strip of size L . Explicitly,the indices of the matrix T L are sequences of L up/down arrows. T L can itself be expressed as aproduct of matrices which encode the vertex weights; in the case of integrable models, we usually enote this matrix by the letter R :(2.1) R = →↑ →↓ ←↑ ←↓→↑ a →↓ b c ←↑ c b ←↓ a Then we have(2.2) T L = tr ( R L · · · R R ) = · · · · · · where R ij means the matrix R acting on the tensor product of i th and j th spaces, and 0 is anadditional auxiliary space encoding the horizontal edges, as on the picture (note that the trace ison the auxiliary space and graphically means that the horizontal line reconnects with itself). Onthe picture “time” flows upwards and to the right.The introduction of a vertical electric field amounts to multiplying the transfer matrix by anoperator which commutes with it, of the form e E y Σ z (Σ z being the number of up arrows minusthe number of down arrows). More interestingly, adding a horizontal field amounts to twistingthe periodic transfer matrix: indeed, all the horizontal fields, using conservation of arrows at eachvertex, can be moved to a single site, so that the transfer matrix becomes, up to conjugation by e E x Σ z ,(2.3) T L = tr ( R L · · · R R Ω)where the twist Ω acts on the auxiliary space and is of the form Ω = e LE x σ z .2.2. Integrability. Properties of the R -matrix. Let us now introduce the following parametrization of the weights: a = q x − q − x − b = x − x − c = q − q − (2.4) x , q are enough to parametrize them up to global scaling. Instead of q one often uses∆ = a + b − c ab = q + q − q or ∆ are fixed whereas x is a variable parameter, called spectral parameter. It can bethought itself as a ratio of two spectral parameters attached to the lines crossing at the vertex.The matrix R ( x ) then satisfies the following remarkable identity: (Yang–Baxter equation) R ( x /x ) R ( x /x ) R ( x /x ) = R ( x /x ) R ( x /x ) R ( x /x ) This is formally the same equation that is satisfied by S matrices in an integrable field theory(field theory with factorized scattering, i.e. such that every S matrix is a product of two-body S matrices). AFF F a/c b/c ∆ = −∞ ∆ = − ∆ = ∆ = Figure 10. Phase diagram of the six-vertex model.The R -matrix also satisfies the unitarity equation: R ( x ) R ( x − ) = ( q x − q − x − )( q x − − q − x ) 112 2 with x = x /x . The scalar function could of course be absorbed by appropriate normalization of R .2.2.2. Commuting transfer matrices. Consider now the transfer matrix as a function of the spectralparameter x , possibly with a twist:(2.5) T L ( x ) = tr ( R L ( x ) · · · R ( x ) R ( x )Ω)Then using the Yang–Baxter equation repeatedly one obtains the relation[ T L ( x ) , T L ( x ′ )] = 0We thus have an infinite family of commuting operators. In practice, for a finite chain T L ( x ) is aLaurent polynomial of x so there is a finite number of independent operators.Note that we could have used the more general inhomogeneous transfer matrix T L ( x ; x , . . . , x L ) = tr ( R L ( y L /x ) · · · R ( y /x ) R ( y /x )Ω)where now we have spectral parameters y i attached to each vertical line i and one more parameter x attached to the auxiliary line. Then the same commutation relations hold for fixed y i andvariable x .As is well-known, the commutation of the transfer matrices is only one relation in the Yang–Baxter algebra generated by the so-called RTT relations. The latter lead to an exact solution ofthe model using Algebraic Bethe Ansatz [31].2.3. Phase diagram. The phase diagram of the six-vertex model in the absence of electric fieldis discussed in great detail in chapter 8 of [4]. It can be deduced from the exact solution of themodel using Bethe Ansatz after taking the thermodynamic limit. The physical properties of thesystem depend only on ∆ = ( q + q − ) / x playing the role of a lattice anisotropy parameter. Wedistinguish three phases, see Fig. 10: ror (a)(b)(c) Figure 11. Correspondence between (a) (∆ = 0) six-vertex (b) NILPs and (c)domino tilings.(1) ∆ ≥ 1: the ferroelectric phase. This phase is non-critical. Furthermore, there are no localdegrees of freedom: the system is frozen in regions filled with one of the vertices of type a or b (i.e. all arrows aligned), and no local changes (that respect arrow conservation) arepossible.(2) ∆ < − 1: the anti-ferroelectric phase. This phase is non-critical. This time there is a finitecorrelation length. The ground state of the transfer matrix corresponds to a state with zeropolarization (in the limit ∆ → −∞ , it is simply an alternation of up and down arrows).(3) − ≤ ∆ < 1: the disordered phase. This phase is critical. It possesses a continuum limitwith conformal symmetry, and this limiting infra-red Conformal Field Theory is well-known:it is simply the c = 1 theory of a free boson on a circle with radius R given by R = − γ/π ) ,∆ = − cos γ , 0 < γ < π .The phase diagram in the presence of an electric field is more complicated, though the basicdivision into the three phases above remains. See [95, 84] for a description. Free fermion point. Inside the disordered phase, there is a special point ∆ = 0. We providevarious representations of the six-vertex model which display the free fermionic behavior of thisregion of parameter space.2.4.1. NILP representation. It is tempting to try to interpret the “north-east going paths” of Fig. 8as Non-Intersecting Lattice Paths. The problem is that they can touch at vertices. One way to fixit is to consider the slightly modified paths of Fig. 11(b) The rule is to replace each vertex of (a)with the corresponding dotted square of (b) and then patch together the latter to form the paths. Note that the correspondence is no longer one-to-one: each vertex of type c corresponds to twopossible local paths. Note that the discussion of the phase diagram in [104] is incomplete. Going from (a) to (b) amounts to combining the equivalences of [47] and [105]. he directed graph of the NILPs is the basic pattern α βγδǫ repeated, with paths movingupwards and to the right, and with weights indicated on the edges. Comparing the weights we getthe relations a = αβδ a = 1 b = βǫ b = αγc = δ + ǫγ c = αβ Combining these we find that a a + b b − c c = 0, so the correspondence only makes sense at∆ = 0 (and there are really only 4 parameters and not 5 as one might naively assume).2.4.2. Domino tilings. There is also a prescription to turn six-vertex configurations into dominotilings that is illustrated on Fig. 11(c) [105]. As already mentioned, going from (b) to (c) is nothingbut a slightly modified version of the bijection of [47] between NILPs and domino tilings.In order to understand the correspondence of Boltzmann weights, note that patching togetherthe pictures of Fig. 11(c) produces dominoes that span three dotted squares, for exampleIn particular, one half of the domino is contained inside one square. This allows to classify dominoesinto four kinds, depending on which half of the square it occupies (these are called north-, west-,south-, and east-going in [47]). Going back to Fig. 11(c), we conclude that a , a , b , b can beconsidered as the Boltzmann weights of the four kinds of dominoes. Furthermore, we have therelations c = a a + b b c = 1from which we derive as expected a a + b b − c c = 0.Just as plane partitions are dimers on the honeycomb lattice, domino tilings can be consideredequivalently as dimers on the square lattice.2.4.3. Free fermionic five-vertex model. The general five-vertex model is obtained by sending one ofthe a or b weights to zero while all other weights remain finite; in other words, one simply forbidsone of the 6 types of vertices. For a discussion of the general five-vertex model , see for example [83]and in particular its appendix A. In the first part of this section, we choose to send both horizontaland vertical electric fields to minus infinity and a to zero, in such a way that a becomes zero. In therepresentation in terms of north-east going paths, this amounts to forbidding crossings; however,these paths in general interact when they are close to each other. The paths become NILPs (i.e.they only interact through the Pauli principle) only if their weights are products over the edges,which implies that b b = c c . This leads us back to the model of section 2.4.1, but with δ sent tozero: what we get this way is the free fermionic five-vertex model, first discussed in [101].If δ = 0 the NILPs of Fig. 11(b) simply live on a regular square lattice, and of course at thisstage we recognize the transfer matrix discussed in section 1.2.4, and illustrated on Fig. 1 (plain igure 12. From the five-vertex model to dimers or plane partitions. Figure 13. From the five-vertex model to dimers or plane partitions, dual version.lines). In section 1.3 on plane partitions, it was also identified with the transfer matrix of lozengetilings. To complete the circle of equivalences, we show on Fig. 12 how to go from NILPs to eitherdimers on the honeycomb lattice or lozenge tilings, following Reshetikhin [93].There is a second case which is worth mentioning (if only because it will reappear in section5.2.5): suppose instead that we send b to zero. This time the north-east going paths cannot gostraight east any more. In this case it is natural to redraw all north-east moves with a right turnas straight lines (not just south-side-goes to east but also west-side-goes-to-north), and this waywe recognize the dashed lines of Fig. 1, with a slight modification: the whole picture is distorted insuch a way that each path moves one step further to the right (so that north-west becomes north,and north becomes north-east). If we want these paths to be NILPs, we reproduce the weights of2.4.1 with γ = 0. Finally the correspondence to lozenge tilings/dimers is illustrated on Fig. 13.Note a difference between the models of lozenge tilings corresponding to these two versions ofthe free fermionic five-vertex models: the vertical spectral parameters flow north-east in the firstpicture, whereas they flow north-west in the second picture. Ultimately, this is related to twopossible inhomogeneous versions of Schur functions (double vs dual [double] Schur functions in thelanguage of [80]). See also the recent work [109] where these lozenge tilings are embedded in a moregeneral square-triangle-rhombus tiling model.2.5. Domain Wall Boundary Conditions. Domain Wall Boundary Conditions (DWBC) werespecial boundary conditions which were originally introduced in order to study correlation functionsof the six-vertex model [60]. However they are also interesting in their own right. igure 14. An example of configuration with Domain Wall Boundary Conditions.2.5.1. Definition. DWBC are defined on a n × n square grid: all the external edges of the grid arefixed according to the rule that vertical ones are outgoing and horizontal ones are incoming. Anexample is given on Fig. 14.To each horizontal (resp. vertical) line one associates a spectral parameter x i (resp. y j ). Thepartition function is thus: Z n ( x , . . . , x n ; y , . . . .y n ) = X configurations n Y i,j =1 w ( y j /x i )where w = a, b, c depending on the type of vertex (cf (2.4)). Here we do not allow any electric fieldfor the simple reason that with DWBC (as with any fixed boundary conditions), using the sametype of arguments as in the previous section, one can push the effect of the field to the boundary,where it only contributes a constant to the partition function. Remark: the (one-to-many) correspondence of section 2.4.2 sends DWBC six-vertex configura-tions to domino tilings of the Aztec diamond [46].2.5.2. Korepin’s recurrence relations. In [60], a way to compute Z n inductively was proposed. It isbased on the following properties: • Z = q − q − . • Z n ( x , . . . , x n ; y , . . . .y n ) is a symmetric function of the { x i } and of the { y i } (separately).This is a consequence of repeated application of the Yang–Baxter equation (or equivalentlyof one of the components of the so-called RTT relations):( q y i +1 /y i − q − y i /y i +1 ) Z n ( . . . , y i , y i +1 , . . . ) = ( q y i +1 /y i − q − y i /y i +1 ) y i y i +1 y Figure 15. Graphical proof of the recursion relation.= y i y i +1 = y i y i +1 = · · · = y i y i +1 = ( q y i +1 /y i − q − y i /y i +1 ) y i y i +1 = ( q y i +1 /y i − q − y i /y i +1 ) Z n ( . . . , y i +1 , y i , . . . )and similarly for the x i . • Z n multiplied by x n − i (resp. y n − i ) is a polynomial of degree at most n − x i (resp. y i ). This is because (i) each variable say x i appears only on row i (ii) a , b arelinear combinations of x − i , x i and c is a constant and (iii) there is at least one vertex oftype c on each row/column. • The Z n obey the following recursion relation:(2.6) Z n ( x , . . . , x n ; y = x , . . . , y n )= ( q − q − ) n Y i =2 ( q x /x i − q − x i /x ) n Y j =2 ( q y j /x − q − x /y j ) Z n − ( x , . . . , x n ; y , . . . , y n )Since y = x implies b ( y /x ) = 0, by inspection all configurations with non-zero weightsare of the form shown on Fig. 15. This results in the identity.Note that by the symmetry property, Eq. (2.6) fixes Z n at n distinct values of y : x i , i = 1 , . . . , n .Since Z n is of degree n − y , it is entirely determined by it. .5.3. Izergin’s formula. Remarkably, there is a closed expression for Z n due to Izergin [40, 39]. Itis a determinant formula:(2.7) Z n = Q ni,j =1 ( x j /y i − y i /x j )( q x j /y i − q − y i /x j ) Q ≤ i The Izergin determinant formula iscurious because it involves a simple determinant, which reminds us of free fermionic models. Andindeed it turns out that it can be written in terms of free fermions, or equivalently that it provides asolution to a hierarchy of classically integrable PDE, in the present case the two-dimensional Todalattice hierarchy. We cannot go in any details here but provide a few remarks.Consider a function of two sets of n variables of the form(2.8) τ n ( X, Y ) = det φ ( X, Y )∆( X )∆( Y )where if X = ( x , . . . , x n ), Y = ( y , . . . , y n ), thendet φ ( X, Y ) = det i,j =1 ,...,n φ ( x i , y j ) ∆( X ) = Y ≤ i 1, we find(2.10) τ n +1 τ n − = τ n ∂∂t ∂∂s τ n − ∂∂t τ n ∂∂s τ n which is a form of the Toda lattice equation .There is another representation which is particular useful for the homogeneous limit. Considerthe Laplace (or Fourier, we are working at a formal level) transform of φ : φ ( x, y ) = Z Z dµ ( a, b ) exa + yb Then one can write τ n ( X, Y ) = 1 n !∆( X )∆( Y ) Z · · · Z n Y i =1 dµ ( a i , b i ) det i,j =1 ,...,n ( ex i a j ) det i,j =1 ,...,n ( ey i b j )This is formally identical to the partition function of a generalized two-matrix model with externalfields for both matrices.Next, let us consider the homogeneous limit τ n ( x, y ) of such a function τ n ( X, Y ) where all x i tendto x and all y i tend to y . Noting that det( e x i a j ) / ∆( X ) ∼ c n e x P i a i ∆( a ) where c n = ( P n − k =1 k !) − (by the usual trick of taking x i = x + iǫ , ǫ → 0) and similarly for the y i , τ n ( x, y ) = c n c n +1 Z · · · Z n Y i =1 dµ ( a i , b i )∆( a ) ex P ni =1 a i ∆( b ) ey P nj =1 b i This is a generalized two-matrix model with linear potentials. With an arbitrary potential, thepartition function of such a model is known to be a tau-function of the two-dimensional Todalattice hierarchy. In fact, we have the following fermionic representation, with notations similar tosection 1: τ n ( x, y ) ∝ h n, n | e P q ≥ xJ + , + yJ − , (cid:16) Z dµ ( a, b ) ψ ⋆ + ( a ) ψ ⋆ − ( b ) (cid:17) n | , i Since we only have here linear potentials i.e. the primary times (the “ t ”), we shall only recoverthe first equation of the hierarchy. Let us do so. First note the determinant formula τ n ( x, y ) = c n det i,j =0 ,...,n − Z Z dµ ( a, b ) a i b j exa + yb = c n det i,j =0 ,...,n − (cid:18) ∂ i ∂x i ∂ j ∂y j φ ( x, y ) (cid:19) Of course the latter form could have been derived directly from (2.8). Next apply to either of theseexpressions the Desnanot–Jacobi determinant identity. More precisely, consider the matrix of size n + 1 and write that its determinant times the determinant of the sub-matrix of size n − n with one row, one column, removed and the other row, other columnremoved (among the last two rows and columns). The result is:(2.11) n τ n +1 τ n − = τ n ∂∂x ∂∂y τ n − ∂∂x τ n ∂∂y τ n (the factor of n takes care of the c n ). inally, in the special case that φ ( x, y ) only depends on x − y , then the previous formulae simplify.The measure dµ ( a, b ) is concentrated on a = − b and we have τ n ( X, Y ) = 1∆( X )∆( Y ) Z · · · Z n Y i =1 dµ ( a i ) det i,j =1 ,...,n ( ex i a j ) det i,j =1 ,...,n ( e − y i a j )which is a generalized one-matrix model with external field. In the homogeneous limit, τ n ( x, y )becomes a function of a single variable t = x − y and we can write(2.12) τ n ( t ) = Z · · · Z n Y i =1 dµ ( a i )∆( a ) et P ni =1 a i which is a one-matrix model with linear potential. With an arbitrary potential, the partitionfunction of such a model is known to be a tau-function of the Toda chain hierarchy. Writing asbefore τ n ( t ) = 1( n !) det i,j =0 ,...,n − Z Z dµ ( a ) a i + j eta = 1( n !) det i,j =0 ,...,n − (cid:18) ∂ i + j ∂t i + j φ ( t ) (cid:19) and applying the Jacobi–Desnanot identity we obtain(2.13) n τ n +1 τ n − = τ n τ ′′ n − τ ′ n which is a form of the Toda chain equation .2.5.5. Thermodynamic limit. In [61], the Toda chain equation (2.13) above was used to derive theasymptotic behavior of the partition function of the six-vertex model with DWBC in two of itsthree phases: ferroelectric and disordered. These are the two phases where we expect the limit n → ∞ to be smooth, as we shall explain. In this case, making the Ansatz that the free energy isextensive, we plug the asymptotic expansion τ n ∼ e − n f into the Toda chain equation and are leftwith a simple differential equation to solve:(2.14) f ′′ = e f In fact, this is the simplest reasonable Ansatz that is compatible with (2.13), so that any solutionof (2.13) with a smooth large n limit will be governed by the differential equation (2.14).This is how the computation of the bulk free energy of the six-vertex model with DWBC for∆ > − Z /n n → ( max( a, b ) ∆ > i a b π/γ cos πt/γ | ∆ | < , q = − e − iγ ( γ > , x = e i ( t + γ ) Note the obvious interpretation of the result in the ferroelectric regime – a frozen phase wherealmost all arrows are aligned.In the anti-ferroelectric phase ∆ < − n limit because thealternation of arrows that is favored in this phase will interact with the boundaries. This statementis made more precise in [105], where matrix model techniques are used to compute the large n limitof (2.12) (which essentially boil down to an appropriate saddle point analysis of it). And indeedone finds that log τ n has an oscillating term of order 1, which explains that the simple differentialequation (2.14) cannot account for the asymptotic behavior.In any case, in all phases one finds a result that is different from that of PBC. The explanationof this phenomenon is that the six-vertex model suffers from a strong dependence on boundaryconditions due to the constraints imposed by arrow conservation. In particular there is no ther-modynamic limit in the usual sense (i.e. independently of boundary conditions). In [104] it was 000 −11 Figure 16. From six-vertex to ASMs.suggested more precisely that the six-vertex model undergoes spatial phase separation, similarly toplane partitions [13] and other dimer models [53]. In other words, even far from the boundary ofthe system, the system loses any translational-invariance and the physical behavior around a givenpoint is a function of the local polarization: as such, the model can have several (possibly, all) ofthe three phases coexisting in different regions. This was motivated by some numerical evidence,as well as by the exact result at the free fermion point ∆ = 0, at which the arctic circle theorem[46] applies: the boundary between ferroelectric and disordered phases is known exactly to be anellipse (a circle for a = b ) tangent to the four sides of the square. So the apparent simplicity of thecomputation of the bulk free energy for DWBC conceals a complicated physical picture.Since then, there has been a considerable amount of work in this area. There has been morenumerical work [1]. Some of the results of [61] have been proven rigorously and extended usingsophisticated machinery in the series of papers [6, 8, 7] by Bleher et al. Finally, the curve separatingphases has been studied in the work of Colomo and Pronko [14, 15], and recently they proposedequations for this curve in the cases a = b , ∆ = ± / / Application: Alternating Sign Matrices. Alternating Sign Matrices are an important class ofobjects in modern combinatorics [77]. They are defined as follows. An Alternating Sign Matrix(ASM) is a square matrix made of 0s, 1s and -1s such that if one ignores 0s, 1s and -1s alternateon each row and column starting and ending with 1s. For example,0 1 0 00 0 1 01 0 − × a = b ),which leaves us with only one parameter c/a , the weight of a ± 1. In fact here we shall consider onlythe pure enumeration problem that is all weights equal. We thus compute ∆ = 1 / q = e iπ/ ,and then x i = q , y j = 1 so that the three weights are w ( x i /y j ) = q − q − .At this stage there are several options. Either one tries to evaluate directly the formula (2.7);since the determinant vanishes in the homogeneous limit where all the x i or y j coincide, this is asomewhat involved computation and is the content of Kuperberg’s paper [62]. here is however a much easier way, discovered independently by Stroganov [98] and Okada[85]. It consists in identifying Z n at q = e iπ/ with a Schur function. Consider the partition λ ( n ) = ( n − , n − , n − , n − , . . . , , λ ( n ) = n − z }| { . . .. . . ... ... ... . . .s λ ( n ) ( z , . . . , z n ) is a polynomial of degree at most n − z i (use (1.24)) and, satisfies thefollowing(2.17) s λ ( n ) ( z , . . . , z j = q − z i , . . . , z n ) = n Y k =1 k = i,j ( z k − q z i ) s λ ( n − ( z , . . . , ˆ z i , . . . , ˆ z j , . . . , z n )where the hat means that these variables are skipped (start from (1.13), find all the zeroes as z j = q z i and then set z i = z j = 0 to find what is left).This looks similar to recursion relations (2.6). After appropriate identification one finds: Z n ( x , . . . , x n ; y , . . . , y n ) (cid:12)(cid:12) q = e iπ/ = ( − n ( n − / ( q − q − ) n n Y i =1 ( q x i y i ) − ( n − s λ ( n ) ( q x , . . . , q x n , y , . . . , y n )Note that Z n possesses at the point q = e iπ/ an enhanced symmetry in the whole set of variables { q x , . . . , q x n , y , . . . , y n } . Finally, setting x i = q − and y j = 1 and remembering that this willgive a weight of ( q − q − ) n to each ASM, one concludes that the number of ASMs is given by A n = 3 − n ( n − / s λ ( n ) (1 , . . . , | {z } n ) = 3 − n ( n − / Y ≤ i Vertices of (a) the CPL model and (b) the FPL model.As a check, one can take n → ∞ in (2.18), and using Stirling’s formula one finds A /n n → √ γ = 2 π/ 3, where we find Z /n n → i/ 8. The two formulaeagree considering Z n = ( √ i/ n A n .3. Loop models and Razumov–Stroganov conjecture Definition of loop models. Loop models are an important class of two-dimensional statis-tical lattice models. They display a broad range of critical range of critical phenomena, and in factmany classical models are equivalent to a loop model. The critical exponents, formulated in thelanguage of loop models, often acquire a simple geometric meaning; and many methods have beenused to study their continuum limit, including the Coulomb Gaz approach [82], Conformal FieldTheory [5] and more recently the Stochastic L¨owner Evolution [65, 66, 67]. Here we are of coursemore interested in their properties on a finite lattice (in relation to combinatorics) and in the useof integrable methods.We shall introduce two classes of loop models on the square lattice, which turn out to be bothclosely related to the six-vertex model. Then we shall discuss a very non-trivial connection betweenthese two loop models (the Razumov–Stroganov conjecture).Let us first discuss common features of the two models. Their configurations consist of loopsliving on the edges of the square lattice. The most important feature is the non-local Boltzmannweight produced by assigning a fugacity of τ to closed loops. Here τ is a real parameter (usuallycalled n , due to the connection with the O ( n ) model – however for various reasons we avoid thisnotation here). This can be supplemented by possible local weights for the various configurationsaround a given vertex, see Fig. 17.3.1.1. Completely Packed Loops. Configurations of Completely Packed Loops (CPL) consist of non-intersecting loops occupying every edge of the square lattice, which produces two possibilities ateach vertex, represented on Fig. 17(a). Besides the weight τ of each closed loop, one can introducea local weight of u for one of the two types of CPL vertices, say NE/SW loops.The model is known to be critical for | τ | < 2, and its continuum limit is described by a theorywith central charge c = 1 − γ π ( π − γ ) , where τ = 2 cos γ , 0 < γ < π .3.1.2. Fully Packed Loops: FPL and FPL models. Configurations of Fully Packed Loops (FPL)consist of non-intersecting loops such that there is exactly one loop at each vertex, which resultsin the six possibilities described on Fig. 17(b). One can then give a weight of τ to each closed loop(FPL model). However, one can do better: noting that the empty edges also form loops (dashedlines on the figure), one can put them on the same footing as occupied edges and assign them afugacity too, say ˜ τ . This more general model is usually called FPL model. ddeven Figure 18. From six-vertex to FPLs.One reason that the FPL is interesting is the following: the FPL model is not integrable for τ = 1 (the very special case τ = 1 is of interest to us and will be considered below). However, theFPL is integrable for τ = ˜ τ , that is if the two types of loops are given equal weights. This wasshown using Coordinate Bethe Ansatz in [21] and then rederived using Algebraic Bethe Ansatz in[43].For generic values of τ , ˜ τ , the Coulomb Gaz approach provides non-rigorous arguments to identifythe continuum field theory, see [41], and allows to compute the central charge to be c = 3 − γ π ( π − γ ) − ˜ γ π ( π − ˜ γ ) , where τ = 2 cos γ and ˜ τ = 2 cos ˜ γ , 0 < γ, ˜ γ < π . In particular the FPL model has centralcharge c = 2 − γ π ( π − γ ) , which is one more than the corresponding CPL model.We shall not discuss the possibility of adding local weights in detail. Let us simply note thateven if we impose rotational invariance of local Boltzmann weights, we can introduce an energycost for 90 degrees turns of the loops, which amounts to giving them a certain amount of bendingrigidity. Such a model was studied numerically in [42].3.2. Equivalence to the six-vertex model and Temperley–Lieb algebra. From FPL to six-vertex. The relation between six-vertex model and FPL model is ratherlimited, so we treat it first. The limitation comes from the fact that one cannot assign an actualweight to the loops, so that we obtain a τ = 1 model (with only local weights). The correspondencebetween configurations is one-to-one: starting from the six-vertex model side, one imposes that atevery vertex, arrows pointing in the same direction should be in the same state (occupied or empty)on the FPL side. This forces us to distinguish odd and even sub-lattices, and leads to the rules ofFig. 18.For rotational invariance of the FPL weights one should have a = b . c/a then plays the role ofrigidity parameter of the loops mentioned in section 3.1.2.The rest of this section is devoted to the equivalence of CPL and six-vertex models.3.2.2. From CPL to six-vertex. An example is shown on Fig. 19.Start from a CPL configuration. The (unoriented) loops carry a weight of τ . A convenient wayto make the latter weight local is to turn unoriented loops into oriented loops : each configuration isnow expanded into 2 configurations with every possible orientation of the loops. The weightof a 90 degrees turn is chosen to be ω ± / , where τ = ω + ω − . + · · · → + · · · Figure 19. From CPLs to six-vertex.Finally we forget about the original loops, retaining only the arrows. We note that the arrowconservation is automatically satisfied around each vertex: we thus obtain one of the six vertexconfigurations. a = = = ub = = = 1 c = + = u ω / + ω − / c = + = ω / + u ω − / Note that if u = 1 all weights become rotationally invariant and a = b , c = c .Finally, one checks that the formula ∆ = − τ / q = − ω ), u playing the roleof spectral parameter. In particular the critical phase | ∆ | < | τ | < Remark : this construction only works in the plane. On the cylinder or on the torus we have aproblem: there are non-contractible loops which according to the prescription above get a weightof 2. This issue will reappear in the section 3.2.4 under the form of the twist. It explains thediscrepancy of central charges between 6-vertex model ( c = 1) and CPL ( c < Link Patterns. In order to understand this equivalence at the level of transfer matrices, oneneeds to introduce an appropriate space of states for the CPL model. We now assume for simplicitythat L is even, L = 2 n .Define a link pattern of size 2 n to be a non-crossing pairing in a disk of 2 n points lying onthe boundary of the disk. Strictly equivalently we can map the disk to the upper half-plane and“flatten” link patterns to pairings inside the upper half-plane of points on its boundary (a line). Weshall switch from one description to the other depending on what is more convenient. The pointsare labelled from 1 to 2 n ; in the half-plane, they are always ordered from left to right, whereas inthe disk the location of 1 must be chosen, after which the labels increase counterclockwise.Denote the set of link patterns of size 2 n by P n . The number of such link patterns is c n = (2 n )! n !( n +1)! ,the so-called Catalan number. xample: in size L = 6, there are 5 link patterns: Periodic Boundary Conditions and twist. Suppose that we consider the CPL model withperiodic boundary conditions in the horizontal direction, with a width of L = 2 n . We can define atransfer matrix with indices living in the set of link patterns P n as follows. Consider appending arow of the CPL model to a link pattern; this way one produces a new link pattern: −→ The transfer matrix T ππ ′ is then the sum of weights of CPL rows such that the pattern π ′ is turnedinto the pattern π . The weights are calculated as follows: first one takes the product over eachplaquette of the local weights; and then one multiplies by τ to the power the number of loops thatwe have created.What is the precise correspondence between the space of link patterns C P n and the space ofspins C L which relates the transfer matrix of the six-vertex model and the newly defined one forthe CPL model?We start from the equivalence described in the section 3.2.2. The basic idea is to orient the loops.So we start from a link pattern and add arrows to each “loop” (pairing of points). Forgetting aboutthe original link pattern we obtain a collection of 2 n up or down arrows, which form a state of the6-vertex model in the transfer matrix formalism. To assign weights it is convenient to think of thepoints as being on a straight line with the loops emerging perpendicularly: this way each loop canonly acquire a weight of ω ± / , depending on whether it is moving to the right of to the left. Forexample, in size L = 2 n = 4, = ω + + + ω − = ω ↑ ↑ ↓ ↓ + ↑ ↓ ↑ ↓ + ↓ ↑ ↓ ↑ + ω − ↓ ↓ ↑ ↑ = ω + + + ω − = ω ↑ ↓ ↑ ↓ + ↑ ↓ ↓ ↑ + ↓ ↑ ↑ ↓ + ω − ↓ ↑ ↓ ↑ There is only one problem with this correspondence: it is not obviously compatible with periodicboundary conditions. We would like to identify a loop from i to j , i < j and a loop from j to i + L , j < i + L . This is only possible if we assume that ↑ i + L = ω ↑ i , ↓ i + L = ω − ↓ i , i.e. we impose twisted oundary conditions on the six-vertex model. In the notations of (2.3) the twist is Ω = ω σ z : itcorresponds to an imaginary electric field.This mapping from the space of link patterns (of dimension c n ) to that of sequences of arrows(of dimension 2 n ) is injective; so that the space of link patterns is isomorphic to a certain subspace C n . The claim, which we shall not prove in detail here but which is a natural consequence of thegeneral formalism is that the transfer matrix (2.3) of the six-vertex model with the twist definedabove leaves invariant this subspace and, once restricted to it, is identical to the transfer matrix ofour loop model up to this isomorphism, the correspondence of weights being the same as in section3.2.2 (in particular, ∆ = − τ / Temperley–Lieb and Hecke algebras. The Temperley–Lieb algebra of size L and with param-eter τ is given by generators e i , i = 1 , . . . , L − 1, and relations:(3.1) e i = τ e i e i e i ± e i = e i e i e j = e j e i | i − j | > Hecke algebra , i.e. the e i satisfy the less restrictive relations(3.2) e i = τ e i e i e i +1 e i − e i = e i +1 e i e i +1 − e i +1 e i e j = e j e i | i − j | > e i ↔ τ − e i . The Hecke algebra isitself a quotient of the braid group algebra : if τ = − ( q + q − ) ( q is thus a free parameter), then the t i = q − / e i + q / satisfy the relations t i t i +1 t i = t i +1 t i t i +1 t i t j = t j t i | i − j | > C ) ⊗ L ofsequences of up/down arrows on which the six-vertex transfer matrix acts. It is given by making e i act on the i th and ( i + 1) st copies of C in the tensor product, with matrix(3.3) e i = − q − q − 00 0 0 0 It may be expressed in terms of Pauli matrices as(3.4) e i = 12 (cid:0) σ xi σ xi +1 + σ yi σ yi +1 + ∆( σ zi σ zi +1 − 1) + h ( σ zi +1 − σ zi ) (cid:1) where h = ( q − q − ) and the σ i are the Pauli matrices at site i ,The second representation of Temperley–Lieb can be defined purely graphically. We now assumeas before that L is even, L = 2 n . In order to define the action of Temperley–Lieb generators e i onthe space of link patterns C P n (vector space with canonical basis the | π i indexed by link patterns),it is simpler to view them graphically as e i = i i +1 ; then the relations of the Temperley–Lieb lgebra, as well as the representation on the space of link patterns, become natural on the picture;for example, we find e = = e = = τ As before, the role of the parameter τ is that each time a closed loop is formed, it can be erased atthe price of a multiplication by τ .The two representations we have just defined are of course related: the transformation of section3.2.4 makes the representation on link patterns a sub-representation of the one on spins.Finally, we need to define affine versions of Temperley–Lieb and Hecke algebras. It is convenientto do so by starting from their non-affine counterparts and adding an extra generator ρ , as well asrelations ρe i = e i +1 ρ , i = 1 , . . . , L − 1, and ρ L = 1. Note that this allows to define a new element e L = ρe L − ρ − = ρ − e ρ such that all defining relations of the algebra become true modulo L . Infact, a more standard approach would be to introduce only e L and not ρ itself. Adding ρ leads to aslightly extended affine Hecke/Temperley–Lieb algebra, which is more convenient for our purposes(see the discussion in [87]).In the link pattern representation, ρ simply rotates link patterns counterclockwise: ρ = In the spin representation, it rotates the factors of the tensor product forward one step and twiststhe last one (that moves to the first position) by Ω. Once again, for these two representations tobe equivalent, one needs Ω to be of the form Ω = ω σ z where ω = − q .As an application, consider the Hamiltonian H L , obtained as the logarithmic derivative of thetransfer matrix T L ( x ) of (2.5) evaluated at x = 1. We find that with periodic boundary conditions,it is simply given up to additive and multiplicative constants by H L = − L X i =1 e i In the spin representation, using (3.4), we recognize in H L the Hamiltonian of the XXZ spin chain (with the so-called ferromagnetic sign convention). So the Hamiltonian of the loop model, whichhas the same form, is equivalent to the XXZ spin chain Hamiltonian, but with twisted periodicboundary conditions, which in terms of Pauli matrices means that σ ± L +1 = ω ± σ ± . ... . . . Figure 20. The CPL model on a cylinder.3.3. Some boundary observables for loop models. Here we consider the CPL model at τ = 1with some specific boundary conditions which will play an important role since the observableswe shall compute live at the boundary. Several geometries are possible and lead to interestingcombinatorial results [17], but here we only consider the case of a cylinder.3.3.1. Loop model on the cylinder. We consider the model of Completely Packed Loops (CPL) on asemi-infinite cylinder with a finite even number of sites L = 2 n around the cylinder, see Fig. 20. Itit convenient to draw the dual square lattice of that of the vertices, so that the cylinder is dividedinto plaquettes . Each plaquette can contain one of the two drawings and .We furthermore set τ = 1 (or q = e πi/ ), that is we do not put any weights on the loops. Thereare no more non-local weights, and in fact plaquettes are independent from each other. So wecan reformulate this model as a purely probabilistic model, in which one draws independently atrandom each plaquette, with say probability p for and 1 − p for .Finally, we define the observables we are interested in. We consider the connectivity of theboundary points, i.e. the endpoints of loops (which are in this case not loops but paths) lyingon the the bottom circle. We encode them into link patterns (see section 3.2.3). In the presentcontext, they can be visualized as follows. Project the cylinder onto a disk in such a way thatthe boundaries coincide and the infinity is somewhere inside the disk. Remove all loops except theboundary paths. Up to deformation of these resulting paths, what one obtains is a link pattern.The probabilities of occurrence of the various link patterns can be encoded as one vector with c n entries: Ψ L = X π ∈ P n Ψ π | π i where P n is the set of link patterns of size 2 n and Ψ π is the probability of link pattern π .3.3.2. Markov process on link patterns. We now show that Ψ L can be reinterpreted as the steadystate of a Markov process on link patterns. This is easily understood by considering a transfermatrix formulation of the model. As in section 3.2.4, let us introduce the transfer matrix: itcorresponds here to creating one extra row to the semi-infinite cylinder, and encoding not theactual plaquettes but the effect of the new plaquettes on the connectivity of the endpoints. Thatis, T π,π ′ ( p ) is the probability that starting from a configuration of the cylinder whose endpoints are onnected via the link pattern π ′ and adding a row of plaquettes, one obtains a new configurationwhose endpoints are connected via the link pattern π . This form a c n × c n matrix T L ( p ).This transfer matrix is actually stochastic in the sense that(3.5) X π ∈ P n T π,π ′ ( p ) = 1 ∀ π ′ which expresses the conservation of probability. This is of course a special feature of the transfermatrix at τ = 1. Note that (3.5) says that T L ( p ) T has eigenvector (1 , . . . , 1) with eigenvalue 1.The matrix T L ( p ) has non-negative entries; it is easy to show that it is primitive (the entries of T L ( p ) n are positive). These are the hypotheses of the Perron–Frobenius theorem. Therefore, T L ( p )possesses a unique eigenvector Ψ L with positive entries; the corresponding eigenvalue is positiveand is larger in modulus than all other eigenvalues. Now the theorem also applies to T L ( p ) T andby uniqueness we conclude that the largest eigenvalue of T L ( p ) and of T L ( p ) T is 1. In conclusion,we find that the eigenvector with positive entries of T L ( p ), which with a bit of foresight we call Ψ L again, satisfies(3.6) T L ( p )Ψ L = Ψ L (In fact the whole reasoning in the previous paragraph is completely general and applies to anyMarkov process, Ψ L being up to normalization the steady state of the Markov process defined by T L ( p ).)Two more observations are needed. Firstly, (3.6) is clearly satisfied by the vector of probabilitiesthat we defined in the previous paragraph (the semi-infinite cylinder being invariant by addition ofone extra row); it is in fact defined uniquely up to normalization by (3.6). This explains that wehave used the same notation.Secondly, Ψ L is in fact independent of p . The easiest way to see this is to note that p nowplays the role of spectral parameter (explicitly, with the conventions of the six-vertex R -matrix, p = q x − q − x − q x − − q − x ). In particular we conclude that, as in the six-vertex model, [ T L ( p ) , T L ( p ′ )] = 0, sothat these transfer matrices have a common Perron–Frobenius eigenvector.3.3.3. Properties of the steady state: some empirical observations. We begin with an example insize L = 2 n = 8. By brute force diagonalization of the stochastic matrix T L one obtains the vectorΨ L of probabilities:Ψ = 142 + + + + 342 + + + + + + + 742 + We recognize some of our favorite numbers A n , namely 7 and 42.In fact, Batchelor, de Gier and Nienhuis [3] conjectured the following properties for all systemsizes L = 2 n :(1) The smallest probability is 1 /A n , and corresponds to link patterns with all parallel pairings.(2) All probabilities are integer multiples of the smallest probability.(3) The largest probability is A n − /A n , and correspond to link patterns which pair nearestneighbors.By now all these properties have been proven [23, 28], as will be discussed in section 4.3.3.4. The general conjecture. A question however remains: according to property 2 above, if onemultiplies the probabilities by A n , we obtain a collection of integers. The smallest one is 1 and thelargest one is A n − , but what can we say about the other ones?Recall that A n also counts the number of six-vertex model configurations with DWBC. Further-more, we showed that there is a one-to-one correspondence between six-vertex model configurationsand FPL configurations (cf section 3.2.1). In this correspondence, the DWBC become alternatingoccupied and empty external edges for FPL configurations (in short, FPLs). Let us now drawexplicitly the 42 FPLs of size 4 × nterestingly, we find that the reformulation in terms of FPLs allows to introduce once again anotion of connectivity. Indeed, there are 2 n occupied edges on the exterior square and they arepaired by the FPL. We can therefore count separately FPLs with a given link pattern π ; let usdenote this by A π . The Razumov–Stroganov conjecture [90] then states thatΨ π = A π A n thus relating two different models of loops (CPL and FPL) with completely different boundaryconditions. And even though both models are equivalent to the six-vertex model, the values of ∆are also different (they differ by a sign).The Razumov–Stroganov conjecture remains open, though some special cases have been proved,see 4.3.2 below.The relation to the conjectured properties of the previous section is as follows. It is easy to showthat if π is a link pattern with all pairings parallel, then there exists a unique FPL configurationwith connectivity π . Thus the RS conjecture implies property (1). Furthermore, since all A π areinteger, it obviously implies property (2). Property (3) however remains non-trivial, since evenassuming the RS conjecture it amounts to saying that A π = A n − in the case of the two linkpatterns π that pair nearest neighbors, which has not been proven.4. The quantum Knizhnik–Zamolodchikov equation We now introduce a new equation whose solution will roughly correspond to a double gener-alization of the ground state eigenvector Ψ L of the loop model introduced above: (i) it containsinhomogeneities and (ii) it is a continuation of the original eigenvector to an arbitrary value of q ,the original value being q = e πi/ .4.1. Basics. The q KZ system. Let L = 2 n be an even positive integer. Introduce once again the R -matrix,but this time rotated 45 degrees and which acts a bit differently than before. Namely, it acts onthe vector space C P n spanned by link patterns, in the following way:(4.1) ˇ R i ( z ) = φ ( z ) ( q − − qz ) + (1 − z ) e i q − z − q i = 1 , . . . , L − e i . φ ( z ) is a scalar function that we do not needto specify explicitly here, see section 4.1.2 for a discussion. Redrawing slightly the latter as e i = and similarly the identity as 1 = , we recognize the two (rotated) CPL plaquettes.Note that in this section, it is convenient to use spectral parameters z that are the squares of our oldspectral parameters x . Indeed, using the equivalence to the six-vertex model described in section3.2.4, which amounts to the representation (3.3) for the Temperley–Lieb generators (acting on the i th and ( i + 1) st spaces), we essentially recover the R -matrix of the six-vertex model, cf (2.1,2.4),after the change of variables z = x :ˇ R ( x ) = φ ( x ) q x − − q − x q x − q − x − q − q − ) x − x − x − x − x − ( q − q − ) x 00 0 0 q x − q − x − provided one performs the following transformation: ˇ R ( z ) ∝ P x κ/ R ( x ) x − κ/ where P permutesthe factors of the tensor product, and κ = diag(0 , , − , R i ( z ) ˇ R i +1 ( z w ) ˇ R i ( w ) = ˇ R i +1 ( w ) ˇ R i ( z w ) ˇ R i ( z ) i = 1 , . . . , L − φ ( z ) φ (1 /z ) = 1, so that ˇ R i satisfies the unitary equation:ˇ R i ( z ) ˇ R i (1 /z ) = 1Consider now the following system of equations for Ψ L , a vector-valued function of the z , . . . , z L , q, q − :( i = 1 , . . . , L − 1) ˇ R i ( z i +1 /z i )Ψ L ( z , . . . , z L ) = Ψ L ( z , . . . , z i +1 , z i , . . . , z L )(4.2) ρ − Ψ L ( z , . . . , z L ) = κ Ψ L ( z , . . . , z L , s z )(4.3)where it is recalled that ρ rotates link patterns counterclockwise, and κ is a constant that is neededfor homogeneity. s is a parameter of the equation: if one sets s = q k + ℓ ) with k = 2 (technically,this is the dual Coxeter number of the underlying quantum group U q ( d sl (2))), then ℓ is called the level of the q KZ equation. (4.2) is the exchange equation, which is ubiquitous in integrable models. (4.3)has something to do with our (periodic) boundary conditions, or equivalently with an appropriateaffinization of the underlying algebra.This system of equations first appeared in [96] in the study of form factors in integrable models.It is not what is usually called the quantum Knizhnik–Zamolodchikov ( q KZ) equation; the latterwas introduced in the seminal paper [35] as a q -deformation of the Knizhnik–Zamolodchikov (KZ)equation ( q KZ is to quantum affine algebras what KZ is to affine algebras). The usual q KZ equationinvolves the operators S i , which S i can be defined pictorially as S i = · · · i · · · where the empty box is just the “face” graphical representation of the R -matrix (dual to the“vertex” representation we have used for the six-vertex model):= q − − q zq − z − q + 1 − zq − z − q and the spectral parameters z to be used in S i are as follows: for the box numbered j , z j /z i if j > i or z j / ( s z i ) if j < i . Loosely, S i is the “scattering matrix for the i th particle”. We use ρ − in the l.h.s. because ( ρ − Ψ L ) π = Ψ ρ ( π ) ! lternatively, S i can be expressed as a product of ˇ R i and ρ :(4.4) S i = ˇ R i − ( z i − / ( s z i )) · · · ˇ R ( z / ( s z i ) ˇ R ( z / ( s z i )) ρ ˇ R L − ( z L /z i ) · · · ˇ R i +1 ( z i +2 /z i ) ˇ R i ( z i +1 /z i )The quantum Knizhnik–Zamolodchikov equation can then be written(4.5) S i ( z , . . . , z L )Ψ L ( z , . . . , z i , . . . , z L ) = Ψ L ( z , . . . , s z i , . . . , z L )It is a simple exercise to check using (4.4) that the system (4.2–4.3) implies the q KZ equation(4.5). Naively, the converse is untrue. However one can show that, up to some linear recombinations,a complete set of (meromorphic) solutions of (4.5) can always be reduced to a complete set ofsolutions of (4.2–4.3), see [11].4.1.2. Normalization of the R -matrix. In the usual setting of the q KZ equation, the normalization φ ( z ) of the R -matrix is chosen such that it satisfies an additional constraint, the crossing symmetry,as well as certain analyticity requirements. There is however a certain freedom in choosing thisnormalization. Indeed, consider a solution Ψ of the system (4.2,4.3) and redefine˜Ψ = Y ≤ i The role of the affine Hecke algebra (and even double affineHecke algebra) was emphasized in the work of Cherednik [12] in relation to Macdonald polynomialsand the quantum Knizhnik–Zamolodchikov equation, and in the work of Pasquier [87] on integrablemodels. In particular, its use in the context of the Razumov–Stroganov conjecture was advocatedby Pasquier [89].Start from the q KZ system (4.2–4.3), and rewrite it in such a way that the action on the finite-dimensional part (on the space of link patterns) is separated from the action on the variables (onthe space of polynomials of L variables). Start with (4.2); after simple rearrangements it becomes(4.6) ( τ − e i )Ψ L = − ( q z i − q − z i +1 ) ∂ i Ψ L where ∂ i ≡ z i +1 − z i ( s i − 1) and s i is the operator that switches variables z i and z i +1 , so that thel.h.s. only acts on the polynomial part of Ψ L , whereas the r.h.s. only acts on link patterns.The operators τ − ˆ e i := − ( q z i − q − z i +1 ) ∂ i acting on the space of polynomials form a representa-tion of the Hecke algebra (with parameter τ ); i.e. they satisfy the relations of (3.2). Equivalently,the ˆ e i satisfy them and form a second set of generators. igure 21. From a Young diagram to a link pattern: in this example, from thepartition (2 , , 1) to the pairings (1 , , , , ρ of spectralparameters z z 7→ · · · 7→ z L s z (with the multiplication by κ included). The ˆ e i togetherwith ˆ ρ generate a representation of the affine Hecke algebra. We can write formally: e i Ψ L = ˆ e i Ψ L (4.7) ρ − Ψ L = ˆ ρ Ψ L (4.8)We can now interpret (4.7,4.8) as follows: we have on the one hand a representation of the affineHecke algebra on C P n , the space of link patterns (with generators e i and ρ ); and on the otherhand a representation of the same algebra on C [ z , . . . , z L ], the polynomials of L variables (theˆ e i and ˆ ρ ). Ψ L provides a bridge between these two representations: it is essentially an invariantobject in the tensor product of the two, that is it provides a sub-representation of the space ofpolynomials (explicitly, the span of the Ψ π ) which is isomorphic to the dual of the representationon the link patterns. By dual we mean, defined by composing the anti-isomorphism that sends the e i to themselves and ρ to ρ − (but reverses the order of the products) and transposition. The roleof the anti-isomorphism is most evident when we try to compose equations (4.7,4.8) and find thatˆ e i ˆ e j Ψ = e j e i Ψ (in fact, it is often advocated to make one of the sets of operators act on the right,though we find the notation too cumbersome to use here).So the search for polynomial solutions of (4.2,4.3) is equivalent to finding certain irreduciblesub-representations of the action of affine Hecke on the space of polynomials. Remark : the direct relation between q KZ and representations of an appropriate affine algebraonly works for the A n series of algebras i.e. affine Hecke. For more complicated situations suchas the Birman–Wenzel–Murakami (BWM) algebra, it does not work so well because one cannotseparate the two different actions [88].4.2. Construction of the solution. On general grounds, we only expect polynomial solution forinteger values of the level. Here we shall only need a solution at level 1, that is s = q .4.2.1. q KZ as a triangular system. We shall build this solution in several steps. First, we use a“nice” property of our basis of link patterns, that is, the fact that (4.6) can be written as a triangular linear system in the components of Ψ L . This requires to define an order on link patterns, whichis most conveniently described as follows. Draw once again link patterns as pairings of points ona line and consider the operation described on Fig. 21. It gives a bijection between link patternsof size 2 n and Young diagrams inside the staircase diagram 1 n = ( n − , n − , . . . , n ; it connects i and 2 n + 1 − i (note that it is one of the link patternswith all pairings parallel, which correspond to the smallest probability 1 /A n in the loop model).Consider now the exchange equation (4.6) and write it in components; we find two possibilities: i and i + 1 are not paired. Then we find that (4.6) only involves Ψ π , and implies that q z i − q − z i +1 divides Ψ π , and furthermore Ψ π / ( q z i − q − z i +1 ) is symmetric in the exchangeof z i and z i +1 . • i and i + 1 are paired. Then( q z i − q − z i +1 ) ∂ i Ψ π = X π ′ = π,e i · π ′ = π Ψ π ′ that is it involves the sum over preimages of π by e i viewed as acting on the set of linkpatterns. It turns out there are two types of preimages of a given π : in terms of Youngdiagrams, there is the Young diagram obtained from π by adding one box at i, i + 1 (whichis always possible unless π is the largest element); and there are other Young diagrams thatare included in π . So we can write the equation(4.9) Ψ π +one box at ( i,i +1) = ( q z i − q − z i +1 ) ∂ i Ψ π − X π ′ ⊂ πe i · π ′ = π Ψ π ′ which has the desired triangular structure and allows to build the Ψ π one by one by addingboxes to the corresponding Young diagram. However there is no equation for Ψ n . In factthis triangular system can be explicitly solved [54] (see also [19]) in the sense that every Ψ π can be written as a series of operators acting on Ψ n . We shall not use this explicit solutionhere.From the discussion above, we find that all we need is to fix Ψ n . We use the following simpleobservation, which generalizes the first case in the dichotomy above: If there are no pairings between points i, i +1 , . . . , j in π , then Q i ≤ p 1) (up to multiplication by a constant). It remains however a non-trivial fact that with sucha choice of Ψ n , the construction above is consistent (i.e. independent of the order in which oneadds boxes), and that (4.3) is satisfied, with s = q . This is the subject of the next section. Example: we find in size L = 6Ψ = − (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) q Ψ = (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) q ( z z z q + z z z q − z z z q − z z z q − z z z q − z z z q + z z z q + z z z q + z z z q + z z z q − z z z − z z z )Ψ = − (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) q = − (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) q Ψ = (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) (cid:0) q z − z (cid:1) q ( z z z q − z z z q − z z z q − z z z q − z z z q − z z z q + z z z q + z z z q + z z z q + z z z q + z z z q − z z z )4.2.2. Consistency and Jucys–Murphy elements. Let us go back to the q KZ system (4.7,4.8). Inorder to show that this system of equations is consistent, one must simply simply check that oneobtains the same representation of affine Hecke on both sides (up to duality). Thus, we shallnow check that the center has the same value in both representations. This will fix κ and moreimportantly s .Let us first define a new set of elements of the affine Hecke algebra (Jucys–Murphy elements),namely y i = t i · · · t L − ρ − t − · · · t − i − i = 1 , . . . , L For this section we shall use “knot-theoretic” drawings to explain the equations we write. It is easyto see that if the t i = q − / e i + q / are crossings in the knot theoretic sense (this definition beingitself the skein relation for the Jones polynomial), then the y i have an equally simple description: t i = i i + 1 y i = i where the second picture is embedded in a strip that is periodic in the horizontal direction. Thenit becomes graphically clear that the y i commute: y i y j = ii jj = = y j y i One can also show that the symmetric polynomials of the y ′ i = q − i y i form the center of theaffine Hecke algebra. Thus, assuming that we have an irreducible representation such that we candiagonalize the action of the y ′ i , the (unordered) set of their L eigenvalues is independent of theeigenvector and characterizes the representation.Let us now apply this to our two representations. First, let us make the y i act on link patterns.They have an obvious common eigenvector: the link pattern 1 n which pairs neighbors 2 i − This extra factor of q i comes from our normalization of the t i . In the Hecke algebra they have no preferrednormalization, and one usually chooses it to make this factor disappear, but in Temperley–Lieb there is one (whichmakes the skein relation rotationally invariant), that we have chosen here, and it is unfortunately different. i , i = 1 , . . . , n . Indeed, we find: y i n = i = − q − / i.e. that the action of y i is equivalent to a Reidemeister move I (with negative orientation for i even and positive orientation for i odd). A move I multiplies the Jones polynomial by − q − / (thisamounts to the calculation t i e i = − q − / e i ), so that we obtain(4.11) y i n = − q ( − i / n On the polynomial side, define similarly ˆ y i = ˆ t − i − · · · ˆ t − ˆ ρ ˆ t L − · · · ˆ t i , where ˆ t i = q − / ˆ e i + q − / .One can show that there is an order on monomials (see [89]) such that the operators ˆ y i are uppertriangular. Thus it suffices to evaluate their diagonal elements in the basis on monomials. Theaction of ˆ y i is most easily computed on monomials Q Li =1 z λ i i with non-increasing powers λ i +1 ≤ λ i .In this case we find(4.12) ˆ y i L Y i =1 z λ i i = − κs λ i q n + − i ) L Y i =1 z λ i i + lower termsComparing the two expressions (4.11) and (4.12), we find that s = q and λ = ( λ i ) is, up to aglobal shift, equal to λ ( n ) = ( n − , n − , . . . , , , , n ( n − κ = q − n − .The latter computation looks very similar to a computation in nonsymmetric Macdonald poly-nomials [73]. This is no coincidence. In fact the eigenvectors of ˆ y i are exactly nonsymmetricMacdonald polynomials, see [50]. If we restrict ourselves to leading terms Q Li =1 z λ i ( π ) i which are inbijection with link patterns by labelling closings/openings in decreasing order, e.g. π = λ ( π ) = (3 , , , , , , , π , even though they are distinct from the Ψ π – there isa triangular change of basis, so that only Ψ n is itself a nonsymmetric Macdonald polynomial.4.2.3. Wheel condition. In [89], a particularly attractive characterization of the span of the Ψ π ( z , . . . , z L ), π ∈ P n , is obtained. It will give an independent check of some of the results of the last section, aswe shall see.Consider the following subspace of homogeneous polynomials:(4.13) M n = { f ( z , . . . , z n ) , deg z i f ≤ n − f ( . . . , z, . . . , q z, . . . , q z, . . . ) = 0 } that is polynomials that vanish as soon as an ordered triplet of variables form the sequence 1 , q , q .This vanishing condition is a so-called wheel condition , see [49] for more general ones. We havealso imposed a bound of n − han what is really needed: imposing on the (total) degree deg f ≤ n ( n − 1) would suffice, but thiscondition is less convenient for our purposes. It is easy to check that Ψ n ∈ M n ; more interestingly, the action of τ − ˆ e i = − ( q z i − q − z i +1 ) ∂ i preserves M n (check the vanishing property by discussing separately the three cases: (i) neither i or i + 1 are in the triplet: trivial; (ii) both i and i + 1 are in the triplet: then the prefactor vanishes;and (iii) one of them is: then use the property for the original triplet and the one with i replacedwith i + 1). Since the Ψ π can be built from Ψ n by action of the τ − ˆ e i , one concludes that theyare all in M n . In fact, note that the action of ˆ ρ also leaves M n invariant on condition that s = q ,so that M n is a representation of the affine Hecke algebra.We have an inclusion of the span of the Ψ π inside M n . In order to prove the equality, we shallstate certain useful properties.4.2.4. Recurrence relation and specializations. First, we need the following simple observation. Ac-cording to the dichotomy of section 4.2.1, if π is a link pattern such that the points i and i + 1are unconnected, then q z i − q − z i +1 divides Ψ π , so that Ψ π ( z i +1 = q z i ) = 0. But if i and i + 1are connected (we shall call such a pairing of neighbors a “little arch”), then we can use the wheelcondition to say that Ψ π ( z i +1 = q z i ) vanishes when z j = q z i , j > i + 1, or z j = q − z i , j < i , sothat it is of the form Q i − j =1 ( z i − q z j ) Q nj = i +1 ( z i − q − z j )Φ where by degree consideration Φ canonly depend on the z j with j = i, i + 1. It is already clear that Φ is in M n − and so is a linearcombination of the entries of Ψ L − ; but there is better.Call ϕ i the mapping from P L − to P L which inserts an extra little arch at ( i, i + 1). Then weclaim that(4.14) Ψ π ( . . . , z i +1 = q z i , . . . ) = π Im ϕ i q − ( n − i − Y j =1 ( z i − q z j ) n Y j = i +2 ( q z i − q − z j )Ψ π ′ ( z , . . . , z i − , z i +2 , . . . , z L ) π = ϕ i π ′ A direct proof of this formula is particularly tedious. Let us instead use the following trick. Considerthe following cyclic invariance property: the formulae (4.14) for different values of i = 1 , . . . , L − i = L ; being a little careful of the factors of s = q that appear here and there, we obtain:(4.15) Ψ ϕ L ( π ′ ) ( z = q − z L , . . . , z L ) = q − n − 1) 2 n − Y j =2 ( z L − q z j )Ψ π ′ ( z , . . . , z L − )where ϕ L adds an arch between L and 1 (on top of all other arches in the half-plane picture of linkpatterns). So, to prove (4.14), it is enough to prove (4.15).We shall use the construction of section 4.2.1. Note that the mapping ϕ L is particularly naturalvia the bijection to Young diagrams, since it becomes the embedding of the set of diagrams inside1 n − to that of diagrams inside 1 n . In particular 0 n − is sent to 0 n ; so the first check is compatibilitywith (4.10), i.e. that when π = 0 n in (4.15) we obtain in the r.h.s. the correct prefactors times Ψ n − .The second check is that the formula (4.9) which allows to compute the components of Ψ L , when In fact, one can show that having a partial degree greater than n − n ( n − A more visual proof of this result is given in [108], where it is noted that since ( i, i + 1) and ( i + 1 , i + 2) cannotbe both pairings, Ψ L ( z i , z i +1 = q z i +1 , z i +2 = q z i ) = 0 and then use (4.2) to permute the arguments and get thegeneral wheel condition. estricted to diagrams inside 1 n − , produces the components of Ψ L − . This is rather obviousgraphically: the action of e i is compatible with the natural inclusion of diagrams (it can be defineddirectly in terms of Young diagrams). Furthermore, in order to remain inside 1 n − , one must neveradd boxes at (1 , 2) or ( L − , L ). So we never affect the variables z , z L and up to a shift of indices i → i − 1, this is just the procedure in size L − 2. This is enough to characterize entirely the r.h.s.of (4.15).Next, we need two more closely related properties: • If f ∈ M n , then(4.16) f ( q − ǫ , . . . , q − ǫ n ) = 0 ∀ ( ǫ i ) Dyck paths increments ⇒ f = 0(a Dyck path increment is a sequence ( ǫ i ) i =1 ,..., n of ± P ni =1 ǫ i = 0 and P ji =1 ǫ i ≥ j ≤ n ). So a polynomial in M n is entirely determined by its values at these c n specializations. • We have the specializations:(4.17)Ψ π ( q − ǫ , . . . , q − ǫ n ) = ( ( − n ( n − / ( q − q − ) n ( n − τ | π | if ǫ i = sign( π ( i ) − i ), i = 1 , . . . , n | π | is the number of boxes in the correspondence with Young diagrams of Fig. 21). In otherwords, these values are the entries in the basis of the Ψ π .The first point is shown explicitly in appendix C of [34] and we shall not reproduce the proof here;the second one is simply an application of the recurrence formula (4.14), by removing little archesone by one (see also [28]).4.2.5. Wheel condition continued. Using the properties of the previous section, we can now concludethat dim M n ≤ c n and that the Ψ π are independent polynomials, so that M n is the span of the Ψ π .One could use more representation-theoretic arguments to prove the equality of M n and of thespan of the Ψ π , for example based on the fact that M n , as a representation of the affine Heckealgebra, is irreducible for generic q (in fact, it corresponds to the rectangular Young diagram with2 rows and n columns), but this is not our philosophy here.The wheel condition has the following physical interpretation (borrowed once again from [89]).Consider the case q = ± 1. Then the q KZ equation (4.5) becomes the ordinary KZ equation at level1. It is most conveniently expressed in the spin basis:3 ∂∂z i Ψ = L X j =1 j = i σ i,j + 1 z i − z j Ψwhere σ i,j exchanges spins i and j . The KZ equation is now in its usual form for correlationfunctions in d sl (2) , except for a trivial change of σ i,j − / σ i,j + 1, which is the same asΨ → Ψ Q i In general, the two problems of diagonalizing the transfermatrix of the CPL loop model and finding solutions of the q KZ system are unrelated. Howeverthere is exactly one value of q where a solution of q KZ does in fact provide an eigenvector of thetransfer matrix (with periodic boundary conditions). This is when the parameter s = 1, whichhere occurs when q = e πi/ (other sixth roots of unity are possible but they are either trivial orgive the same result as the one we picked). In this case note that (4.3) becomes a simple rotationalinvariance condition. Furthermore the real q KZ equation (4.5) becomes an eigenvector equationfor the scattering matrices: S i ( z , . . . , z L )Ψ L ( z , . . . , z L ) = Ψ L ( z , . . . , z L )These scattering matrices do not involve any extra shifts of the spectral parameters, and as iswell-known in Bethe Ansatz, are just specializations of the inhomogeneous transfer matrix. Indeedif we define T L ( z ; z , . . . , z L ) to be simply T L = z z z · · · z L (with periodic boundary conditions), where the box represents the R -matrix evaluated at the ratioof vertical and horizontal spectral parameters, then observe that S i ( z , . . . , z L ) = T L ( z i ; z , . . . , z L ).By a Lagrange interpolation argument, we conclude that T L ( z ; z , . . . , z L )Ψ L ( z , . . . , z L ) = Ψ L ( z , . . . , z L )i.e. Ψ L is up to normalization the steady state of the inhomogeneous Markov process defined by T L ( z ; z , . . . , z L ). In order to recover the original homogeneous Markov process, one simply sets all z i = 1.Note that the reasoning above is valid for any solution of q KZ, not just the one that was discussedin section 4.2.1. But there is no point in considering higher degree polynomial solutions since at q = e πi/ they will simply be the minimal degree of section 4.2.1 times a symmetric polynomial.We now show how to apply the q KZ technology to prove some of the statements formulated insection 3.4.3.1. Proof of the sum rule. As a simple application of the above formalism, we explain howto recover the “sum rule” at q = e πi/ . Let us explain what we mean by that. We start bynoting that the normalization of Ψ is fixed once the value of Ψ n is specified; in particular, if all z i = 1, using (4.10) we find Ψ n = 3 n ( n − / , to be compared with the (conjectured) probability1 /A n associated to 0 n , cf section 3.3.3. In other words, with this normalization, one should have π ∈ P n Ψ π = 3 n ( n − / A n . This is what we are going to show now. In fact, following [23], we aregoing to show more generally the inhomogeneous sum rule that P π ∈ P n Ψ π ( z , . . . , z n ) is equalto the Schur function s λ ( n ) ( z , . . . , z n ), where λ ( n ) is defined in (2.16), that is up to a constantthe partition function of the six-vertex model with DWBC at q = e iπ/ . Instead of the inductivemethod of [23], i.e. the comparison of the recurrence relations of sections 4.2.4 and 2.5.2, we shallhere proceed directly.Define the covector v with entries v π = 1, π ∈ P n . The stochasticity property of a matrix isthe fact that v is left eigenvector with eigenvalue 1; and it is satisfied at τ = 1 by all the e i andhenceforth by ˇ R i ( z ): ve i = v ⇒ v ˇ R i ( z ) = v i = 1 , . . . , L − , q = e πi/ Applying this identity to (4.2), we immediately conclude that v · Ψ = P π ∈ P n Ψ π is a symmetricpolynomial of its arguments z , . . . , z L . P π ∈ P n Ψ π ( z , . . . , z n ) is a symmetric polynomial of degree n ( n − 1) which satisfies the wheelcondition of (4.13). We claim that this defines it uniquely up to normalization (see [108] for asimilar claim in a more general fused model). The simplest proof is to use once again property(4.16) i.e. that polynomials in M n are characterized by the specializations ( q − ǫ , . . . , q − ǫ n ). Butfor a symmetric polynomial, all these specializations reduce to only one. This proves the claim.Next, one notes that s λ ( n ) ( z , . . . , z n ) also satisfies these properties. The degree bound is clearfrom the definition (1.24) of Schur functions; as to the wheel condition, it follows from formula(1.23) by noting that if z k = q z j = q z i , the sub-matrix of the matrix of the numerator withcolumns { i, j, k } is of rank 2 and thus the determinant of the whole matrix vanishes.Finally, to fix the normalization constant, note that thanks to (4.17), all components of Ψvanish at ( z , . . . , z n ) = ( q − , . . . , q − , q, . . . , q ) except Ψ n ( q − , . . . , q − , q, . . . , q ) = 3 n ( n − / ; onthe other hand, using (2.17), we find the same value s λ ( n ) ( q − , . . . , q − , q, . . . , q ) = 3 n ( n − / . Weconclude that X π ∈ P n Ψ π ( z , . . . , z n ) = s λ ( n ) ( z , . . . , z n ) Remark: from the representation-theoretic point of view, v is an invariant element (under affineHecke action) in ( C P n ) ⋆ ; and P π ∈ P n Ψ π is its image in C [ z , . . . , z L ]. This shows that at q = e πi/ ,the representation of affine Hecke on C P n (resp. the span of the Ψ π ) is not irreducible (though itremains indecomposable), since it has a codimension one (resp. dimension one) stable subspace.4.3.2. Case of few little arches. Note a remarkable property of recurrence relation (4.14): it canonly decrease the number of little arches! So, if one considers the subset of link patterns with agiven maximum number of little arches, that subset is closed under these relations.The link patterns that possess only two little arches (which is the minimum) are the link pattern0 n and its rotations, i.e. the n link patterns for which all the pairings are parallel and for which weknow explicitly the components: they are given by the action of ˆ ρ (cyclic rotation of variables plusshift of s ) on Ψ n which is (4.10).These are the smallest components in the homogeneous limit. They correspond via the Razumov–Stroganov conjecture to a single FPL. As such they are clearly the “easiest” components.It is natural to look at what happens when one considers more little arches. Link patternswith three little arches are of the form of Fig. 22. In fact, the case of three little arches was firstinvestigated in [29] on the other side of the Razumov–Stroganov conjecture: there, the problem ofthe enumeration of FPLs with such a connectivity was solved, with the remarkable result that the bc ab c γ q γ a + b q α α b + c q β q β a + c Figure 22. Link patterns with 3 arches.number of FPLs with link pattern ( a, b, c ) is simply equal to the number of plane partitions insidean a × b × c box, that is given by the MacMahon formula (1.27). The proof is bijective and weshall not reproduce it here. Let us just comment on it. The crux of the bijection is the observation,known as “de Gier’s lemma”, that in an FPL with given connectivity many edges are fixed by thesimple requirement of the connectivity of the endpoints. Removing all these fixed edges we obtaina simpler enumeration problem, typically of plane partitions. Now the little arches play a key rolein the sense that they are the ones that limit application of de Gier’s lemma. In other words, themore little arches there are, the fewer fixed edges in FPLs, and therefore the more complicated theenumeration is. Once again, the number of little arches is a measure of complexity.Closing this philosophical parenthesis, let us go back to the q KZ equation and try to computethe entries Ψ a,b,c of Ψ L corresponding to link patterns with three series of a , b , c nested arches.This is performed in [107] for q = e πi/ and we generalize it here to arbitrary q .Up to rotation and use of (4.3), one can always assume that one of the little arches, say theone that is part of the series of a nested arches, is between L and 1. Let us further rename the L variables z i as follows: they become ( q − γ , . . . , q − γ a + b , α , . . . , α b + c , q β , . . . , q β a + c ), see Fig. 22.Then according to the results of section 4.2.1,Ψ a,b,c = Y ≤ i 0) is nothing but the base link pattern 0 n , and in this case we find(4.20) Φ a,b, = a Y i =1 b Y j =1 ( α j − β i )Some explicit formulae for Φ a,b,c are given in [107]. In fact they are “triple Schur functions” in thesense of multi-Schur functions of [63]. bc α α α α α α α β β β β β γ γ γ γ γ γ a bc α α α α α α α β β β β β γ γ γ γ γ γ (a) (b) Figure 23. The three sets of spectral parameters associated to lozenge tilings.From the discussion above, it is natural to try to put non-trivial Boltzmann weights on lozengetilings of a hexagon of sides a , b , c in such a way that their partition function inside an a × b × c box coincides with the inhomogeneous component Φ a,b,c . Such a model is introduced in [107]. Wepoint it out here because we believe this model is of some interest and might deserve further study.The spectral parameters live on the medial lattice, which is the Kagome lattice, see Fig. 23(a).Each lozenge has a weight which is the difference of spectral parameters crossing at the center ofthe lozenge, with the sign convention α − β , β − γ , α − γ . One can show that the integrability of the underlying model on the Kagome lattice implies thatthe partition function is a symmetric function of the { α i } , of the { β i } and of the { γ i } . We choosenot to do so here and refer instead to the appendix of [107] for a “manual” proof of symmetry.Finally, it is easy to see that the partition function of lozenge tilings with such weights satisfiesthe recurrence relation (4.19), as illustrated on Fig. 23(b); and that when c = 0, there is a uniquetiling of the a × b parallelogram, resulting in (4.20). Note the similarity with the properties ofthe partition function of the six-vertex model with DWBC (symmetry and recurrence) found byKorepin.Thus, Φ a,b,c and this partition function coincide. Finally, if α i = 1, β i = q − , γ i = q , using thefact that the number of lozenges of each orientation is fixed and equal to ab , bc , ca , we find:Φ a,b,c = (1 − q − ) ab ( q − − q ) bc (1 − q ) ca N a,b,c = ( q − q − ) ab + bc + ca q a ( c − b ) ( − ca N a,b,c q = e πi/ Ψ a,b,c = q − ( a + c )( a + c − a + b )( a + b − − b ( a + b − c ( a +3 b − c ( − ( a + b )( a + b − c ( b +( c − ( q − q − ) ( b + c )( b + c − + ( c + a )( c + a − + ( a + b )( a + b − Φ a,b,c = ( q − q − ) n ( n − ( − n ( n − N a,b,c q = e πi/ We conclude that Ψ a,b,c / Ψ n = N a,b,c , as expected.The case of four little arches is similar to that of three arches and is sketched in [107]. On the onehand, the recurrence relations are once again enough to determine all Ψ π uniquely. On the other This is only a convention because the total number of lozenges of each orientation is conserved. The choice thatwe made does not respect the Z symmetry of the model but turns out to be convenient. and, the enumeration of FPL configurations with four little arches is doable (after appropriateuse of the so-called Wieland rotation [100]) and can be once again reduced to a lozenge tilingenumeration [30]. So the same strategy applies. Starting with five little arches, however, it fails onboth sides: the recurrence relations are not sufficient to determine the corresponding componentsΨ π , and the enumeration of FPLs becomes non-trivial due to an insufficient number of fixed edges.4.4. Integral formulae. We now want to show that, using the formalism of the q KZ equationallows to prove the properties discussed in 3.3.3, as well as to reconnect the three models thatwe have found in which the same numbers A n appear. A particular useful tool to exploit thesesolutions of q KZ has been introduced in [28, 92]: it consists in writing integral formulae for them.4.4.1. Integral formulae in the spin basis. There are various types of known integral formulae forsolutions of the quantum Knizhnik–Zamolodchikov equation. Here we are only interested in a veryspecific one, which only exists at level 1 and is obtained by ( q -)bosonization, see [45].The following formula is valid in even size L = 2 n :(4.21) Ψ spina ,...,a n − ( z , . . . , z L ) = ( q − q − ) n Y ≤ i 7→ { a < · · · < a n − } = { i : α i = + } .For the case of odd size see [92].4.4.2. The partial change of basis. Next we would like to write similar expressions for the entries ofΨ in the basis of link patterns. Unfortunately this would require to invert the change of basis fromlink patterns to spins that was described in section 3.2.4; and there is no simple explicit formulafor the inverse (this is the famous problem of the computation of Kazhdan–Lusztig polynomials[52]; in this case there is a combinatorial formula [64] but it is not obvious how to combine it withintegral formulae).Instead we shall do here something more modest: we introduce an intermediate basis betweenspins and link patterns. We skip here the details, which can be found in [28] (see in particularthe appendices) and [27]. The bottom line is the introduction of a slight improvement of formula(4.21):(4.22) Ψ a ,...,a n − ( z , . . . , z L ) = Y ≤ i 1) = I · · · I n − Y ℓ =0 du ℓ πi u a ℓ ℓ Y ≤ ℓ 1) = Y ≤ ℓ The integral formulae above allow to compute various linearcombinations of the Ψ π . Here, we concentrate on a single quantity of particular interest (which isconsidered in [28, 20]). Consider the linear combination F n ( t ) = P π ∈ P n c π ( t )Ψ π (1 , . . . , n = 1, and where the coefficient c π ( t ) is most simply expressed in terms of thecorresponding Young diagram by assigning a weight of t or t − to the boxes of even/odd rows inthe complement inside 1 n , see Fig. 24. Three special cases are of particular interest: F (1) is justthe sum P π ∈ P n Ψ π of all link patterns; F (0) is just Ψ n , one of the two largest components; and F ( t ) | t n − , the leading term of F ( t ), is just Ψ ρ (1 n ) , the other largest component.We then claim that P π ∈ P n c π ( t )Ψ π can be naturally expressed in the intermediate basis as P ǫ ,...,ǫ n − ∈{ , } t P ni =1 ǫ i Ψ − ǫ , − ǫ ,..., n +1 − ǫ n − (this is a simple calculation of the corresponding en-tries of the matrix of change of basis), which results in an integral formula. In the homogeneouslimit we obtain F n ( t ) = n − Y ℓ =0 (1 + t u ℓ ) Y ≤ ℓ Weight of a component expressed in terms of the associated Young diagram.In fact one can simply set u = 0 to get(4.23) F n ( t ) = n − Y ℓ =1 (1 + t u ℓ )(1 + τ u ℓ ) Y ≤ ℓ 1. With a bit morework, one can also prove that F n (0) = τ n − F n − (1 /τ ). s a result, at τ = 1, F n (1), the sum of components, is A n , and F n ( t ) | t n − = F n (0), the largestcomponents, are A n − .4.4.4. Refined enumeration. For general t , one can show that F n ( t ) corresponds to a refined weightedenumeration of TSSCPPs, that is to adding extra boundary weights on the TSSCPPs. The detailscan be found in [28].Remarkably enough, if one sets τ = 1, then F n ( t ) also coincides with the refined enumerationof ASMs. For ASMs, the refinement is simple to explain and consists in giving a weight of t i − to an ASM whose 1 in the first row is at column i (which is essentially equivalent to keeping onespectral parameter free in the partition function of the six-vertex model with DWBC and sendingthe others to 1). This strange coincidence was observed in [92] and proved via rather indirectarguments (making use of the Bethe Ansatz calculations of [91]).One can push this idea further: in 1986, Mills, Robbins and Rumsey conjectured that the double refinements of TSSCPPs and ASMs coincide [78]. Again, the double refinement of ASMs is easy toexplain, consisting in weights t i − u n − j for an ASM with a 1 at (1 , i ) and at ( n, j ) (which is equivalentto keeping two say horizontal spectral parameters free). The double refinement of TSSCPPs can beformulated in multiple ways and we refer the reader to appendix A of [34] for details. And in fact,using slight generalizations of the integrals above, one can give a direct proof of this conjecture andturn it into a theorem, as is done in [34].5. Integrability and geometry The goal in this section is to reinterpret the families of polynomials that give rise to a solutionof the q KZ equation in terms of geometric data. For simplicity we shall stick to cohomology (asopposed to K -theory), which means that we only consider rational solutions of the Yang–Baxterequation. In terms of the parameter q , it means taking an appropriate q → ± q → − q → − q KZ of section 4. The thirdone (Brauer loop scheme) in fact contains the first two as special cases and is the most interestingone. But first we need to introduce some technology i.e. an algebraic analogue of equivariantcohomology for affine schemes with a linear group action: multidegrees [76]. Note that in thissection we give almost no proofs and refer to the papers for details.5.1. Multidegrees. Our group will always be a torus T = ( C × ) N . T acts linearly on a com-plex vector space W . To a closed T -invariant sub-scheme X ⊆ W we will assign a polynomialmdeg W X ∈ Sym( T ∗ ) ∼ = Z [ z , . . . , z N ] (here T ∗ is viewed as a lattice inside the dual of the Liealgebra of T ) called the multidegree of X .5.1.1. Definition by induction. This assignment can be computed inductively using the followingproperties (as in [48]):1. If X = W = { } , then mdeg W X = 1.2. If the scheme X has top-dimensional components X i , where m i > X i in X , then mdeg W X = P i m i mdeg W X i . This lets one reduce from the case ofschemes to the case of varieties (reduced irreducible schemes).3. Assume X is a variety, and H is a T -invariant hyperplane in W .(a) If X H , then mdeg W X = mdeg H ( X ∩ H ). b) If X ⊂ H , then mdeg W X = (mdeg H X ) · (the weight of T on W/H ) . One can readily see from these properties that mdeg W X is homogeneous of degree codim W X , andis a positive sum of products of the weights of T on W .5.1.2. Integral formula. We can also reformulate the multidegree as an integral (generalizing theidea that the degree of a projective variety is essentially its volume). To the torus T acting linearlyon the complex vector space W is naturally associated a moment map µ , which is quadratic. Withreasonable assumptions (i.e. that the multigrading associated to the torus action is positive ), onecan map generators of T ∗ to complex numbers in such a way that this quadratic form is positive.Then one can formally write mdeg W X = R X dx e − πµ ( x ) R W dx e − πµ ( x )where it is implied that on both sides, this evaluation map has been applied. More explicitly,suppose that ( x i ) i =1 ,...,n is a set of coordinates on W which are eigenvectors of the torus action,with weights α i ; then mdeg W X = R X dx e − π P ni =1 α i | x i | R W dx e − π P ni =1 α i | x i | = n Y i =1 α i Z X dx e − π P ni =1 α i | x i | where the α i must be evaluated to positive real numbers for the integral to make sense. So, themultidegree is just a Gaussian integral. The subtlety comes from the fact that in general X isa non-trivial variety to integrate on (in particular, it will be singular at the torus fixed point 0,the critical point of the function in the exponential!). In the case where X is a linear subspace of W , given by equations x i = · · · = x i k = 0, which is the only case where the integral is really anordinary Gaussian integral, we compute immediately mdeg W X = α i . . . α i k , as expected.5.2. Matrix Schubert varieties. Matrix Schubert varieties are a slight variation of the classicalSchubert varieties, whose study started Schubert calculus. They are simpler objects to define, beingaffine schemes (i.e. given by a set of polynomial equations in a vector space).5.2.1. Geometric description. Let g = gl ( N, C ), b + = { upper triangular matrices } ⊂ g , b − = { lower triangular matrices } ⊂ g , and similarly we can define the groups G = GL ( N, C ) and B ± = { invertible upper/lower triangular matrices } ⊂ G .The matrix Schubert variety S σ associated to a permutation σ ∈ S N is a closed sub-variety of g defined by the equations:(5.1) S σ = { M ∈ g : rank M i j ≤ rank σ Ti j , i, j = 1 , . . . , N } where M i j is the sub-matrix above and to the left of ( i, j ), and σ T is the transpose of the permutationmatrix of σ .Explicitly, S σ is defined by a set of polynomial equations of the form, determinant of a sub-matrixof M is zero, which express the rank conditions.One can also describe S σ as a group orbit closure:(5.2) S σ = B − σ T B +58 vidently, we have S σ − = ( S σ ) T . Less evidently, the codimension of S σ is the inversion number | σ | of σ . Examples: • σ = 1: then rank σ Ti j = min( i, j ) which is the maximum possible rank of a i × j matrix. Sothere is no rank condition and S σ = g . • σ = σ , the longest permutation, σ ( i ) = N + 1 − i : this time rank σ Ti j = max(0 , i + j − N )and we find that S σ is a linear subspace S σ = { M = ( M ij ) i,j =1 ,...,N : M ij = 0 ∀ i, j, i + j ≤ N } = ⋆ ⋆ · · · ⋆ • All the other matrix Schubert varieties are somewhere in between. For example, if σ =(4132), then σ T = ⇒ S (4132) = (cid:26) M = ( M ij ) i,j =1 ,..., : M = M = M = 0 M M − M M = 0 (cid:27) Note that among the multiple determinant vanishing conditions, one can extract in thiscase four that imply all the others. In general, there are simple graphical rules to determinewhich conditions to keep.There is a torus T = ( C × ) N acting on g , by multiplication on the left and on the right bydiagonal matrices. The corresponding generators of its dual are denoted by x , . . . , x N (left) and y , . . . , y N (right). In fact, since multiplication by a scalar does not see the distinction betweenleft and right, the torus acting is really of dimension 2 N − 1, and this amounts to saying that allmultidegrees we shall consider only depend on differences x i − y j .The S σ are T -invariant; let us define the double Schubert polynomials Ξ σ to be their multidegrees:Ξ σ = mdeg g S σ Note that Ξ σ − ( x , . . . , x N | y , . . . , y N ) = Ξ σ ( y , . . . , y N | x , . . . , x N ), so the two sets of variablesplay symmetric roles.5.2.2. Pipedreams. In [32] (see also [56]), a combinatorial formula for the calculation of doubleSchubert polynomials was provided. It can be described as configurations of a simple statisticallattice model coined in [56] (reduced) “pipedreams”.The pipedreams are made of plaquettes, similarly to loop models of section 3. The two allowedplaquettes are and (where the second one should be considered as two lines actuallycrossing). Furthermore, pipedreams have a specific geometry: they live in a right-angled triangle,as shown on the examples below. On the hypotenuse, the plaquettes are forced to be of the firsttype (and only one half of them is represented).A pipedream is reduced iff no two lines cross more than once. Alternatively, one can think ofputting a non-local weight of zero to “bubbles” formed by two lines crossing twice. We assignweights to reduced pipedreams as follows: a crossing at row/column ( i, j ) has weight x i − y j . We an now state the formula for double Schubert polynomials:Ξ σ ( x , . . . , x N | y , . . . , y N ) = X reduced pipedreams of size N such that point i on vertical axisis sent to point σ ( i ) on horizontal axis i =1 ,...,N (weight of pipedream)A sketch of proof of this formula will be given in the next two sections. Examples: • σ = 1: then only one pipedream contributes, the one with no crossings, of the type 123 2 3 414 ⇒ Ξ = 1 • σ = σ : again, only one pipedream contributes, this time with only crossings: 123 2 3 414 ⇒ Ξ σ = Y i + j ≤ N ( x i − y j )Each factor of x i − y j has the meaning of weight of the equation M ij = 0. • If σ = (4132), there are two (reduced) pipedreams: 123 2 3 414 123 2 3 414 ⇒ Ξ (4132) = ( x − y )( x − y )( x − y )( x + x − y − y )Again, we recognize in each factor the weight of one equation of S (4132) . In general, as longas S σ is a complete intersection (as many defining equations as the codimension), Ξ σ is justthe product of weights of its equations (i.e. linear factors).5.2.3. The nil-Hecke algebra. In [33, 32], double Schubert polynomials are related to a solutionof the Yang–Baxter equation (YBE) based on the nil-Hecke algebra. We shall reformulate thisconnection here in terms of our exchange relation.Define the nil-Hecke algebra by generators t i , i = 1 , . . . , N − 1, and relations t i = 0 t i t i +1 t i = t i +1 t i t i +1 t i t j = t j t i | i − j | > s a vector space, the nil-Hecke algebra is isomorphic to C [ S N ] (the t i being identified with el-ementary transpositions). It has an obvious graphical depiction in which basis elements σ ∈ S N permute lines; the product is the usual product in S N except that when lines cross twice, the resultis zero: (2143)(1342) = = = (2431)(1423)(3124) = = 0(the expressions from right to left correspond to the pictures from bottom to top). The inversionnumber of σ is graphically its number of crossings.Define ˇ R i ( u ) = 1 + u t i which is the associated solution to the rational Yang–Baxter equation, that is with additive spectralparameters: ˇ R i ( u ) ˇ R i +1 ( u + v ) ˇ R i ( v ) = ˇ R i +1 ( v ) ˇ R i ( u + v ) ˇ R i +1 ( u )Note that t i = , and the identity is 1 = , that is 45 degrees rotated versions of ourplaquettes, cf a similar remark in section 4.1.1. As usual, our solution of YBE also satisfies theunitarity equation: ˇ R i ( u ) ˇ R i ( − u ) = 1Now consider the formal object Ξ = P σ ∈S N Ξ σ σ viewed as an element of nil-Hecke. We thenclaim that the following formulae hold:Ξ( x , . . . , x N | y , . . . , y N ) ˇ R i ( x i +1 − x i ) = Ξ( x , . . . , x i +1 , x i , . . . , x N | y , . . . , y N )(5.3) ˇ R i ( y i − y i +1 )Ξ( x , . . . , x N | y , . . . , y N ) = Ξ( x , . . . , x N | y , . . . , y i +1 , y i , . . . , y N )(5.4)which is nothing but exchange relations. Note that if we consider X as a function of the first setof variables x , . . . , x N , the nil-Hecke algebra must act on the right on itself (ultimately, this isbecause of the transposition in (5.1) and (5.2)).These formulae are obvious if one defines Ξ in terms of pipedreams, that is if one writes Ξ as [32]Ξ = x x x x y y y y 2’ 3’ 4’1’1234 = here each plaquette is an R -matrix evaluated at y j − x i (row i , column j ), and each pipedream isinterpreted as an element of nil-Hecke according to the convention of endpoints shown next to theequality. The proof of (5.3) is then the usual argument (cf for example section 2.5.2) of applyingthe ˇ R matrix between rows i and i + 1, moving it across using YBE and checking that it disappearsonce it reaches the hypotenuse; and similarly for (5.4).To conform with the literature, we shall now focus on the first form (5.3). Writing it explicitlyin components, we have to distinguish as usual cases. Let us denote as before s i the permutationof variables x i and x i +1 . • If σ ( i ) < σ ( i + 1) (the lines starting at ( i, i + 1) do not cross): then σ is not the image of t i acting by right multiplication, so that we find(5.5) Ξ σ = s i Ξ σ i.e. Ξ σ is symmetric under exchange of x i and x i +1 . • If σ ( i ) > σ ( i + 1) (the lines starting at ( i, i + 1) cross): this time we find that Ξ σ + ( x i +1 − x i )Ξ σ ′ = s i Ξ σ , where σ ′ is the unique preimage of σ under right multiplication by t i . Thus,(5.6) Ξ σ ′ = ∂ i Ξ σ where ∂ i = x i +1 − x i ( s i − σ ′ satisfies σ ′ ( i ) < σ ′ ( i + 1), so that (5.5)can be deduced from (5.6) (the image of ∂ i is symmetric in x i , x i +1 ).In (5.6), σ ′ has one fewer crossings than σ . It is easy to show this way that, starting from Ξ σ = Q i + j ≤ N ( x i − y j ), where σ is the permutation with the most crossings (the highest inversionnumber), one can compute any Ξ σ by repeated use of (5.6).We now explain what the geometric meaning of (5.6) is. This will prove that the pipedreams docompute the multidegrees of matrix Schubert varieties.5.2.4. The Bott–Samelson construction. Let L i be the subgroup of G consisting of matrices whichare the identity everywhere except in the entries M ab with a, b ∈ { i, i + 1 } , and B ± ,i = B ± ∩ L i . Weuse the following lemma, which is similar to lemma 1 of [58] (see also lemma 8 of [59]): (throughoutthe lemma the sign ± is fixed) Let V be a left L i -module, and let X be a variety in V that is invariant underscaling and under the action of B ± ,i ⊂ L i . Define the map µ : ( L i × X ) /B ± ,i → V that sends classes of pairs ( g, x ) (where B ± ,i acts on the right on L i and on the lefton X ) to g · x . If µ is generically one-to-one, thenmdeg V Im µ = ∓ ∂ i mdeg V X If on the other hand X is L i -invariant, then ∂ i mdeg V X = 0Here we apply the lemma with V = g , L i acting on it by left multiplication, and X = S σ .According to (5.2), X is B − ,i -invariant. Furthermore, if σ ( i ) > σ ( i + 1), one can check that themap µ is generically one-to-one, and the image is precisely S σ ′ where σ = σ ′ t i . Hence (5.6) holds.On the contrary, if σ ( i ) < σ ( i + 1), X is simply L i -invariant, and we conclude that (5.5) holds.5.2.5. Factorial Schur functions. Pipedreams contain non-local information carried by the connec-tivity of the lines, just like the loop models of section 3. It is natural to wonder if there is a “vertex”representation similar to the six-vertex model for pipedreams. As this model is based on a Heckealgebra (which we have considered so far in the regular representation), there are in fact plenty f vertex representations corresponding to various quotients of the Hecke algebra. The simplestone, analogous to the Temperley–Lieb quotient, is as follows. Suppose that σ is a Grassmannianpermutation . By definition, this means that σ has at most one descent, i.e. there is a k such that i = k implies σ ( i ) < σ ( i + 1). Then one can group lines into two subsets depending on whetherthey start on the vertical axis at position i ≤ k or i > k . If we use different colors for them, we geta picture such as Several observations are in order. First, there is indeed no more non-local information in the sensethat the connectivity of the endpoints can be entirely determined from the sequences of colors onthe horizontal axis. Indeed, if the green endpoints are a < · · · < a k and the red endpoints are b < · · · < b n − k , there is only one Grassmannian permutation which is compatible with these colors,namely ( a , . . . , a k , b , . . . , b n − k ). Secondly, there is never any crossing below row k , so Ξ σ does notdepend on x k +1 , . . . , x N . According to (5.5), we also know that it is a symmetric polynomial of x , . . . , x k . Thirdly, there are now five types of plaquettes: the four colorings of the non-crossingplaquette, but only one crossing plaquette (lines of the same color cannot cross each other!). In fact,by identifying red to occupied and green to empty, we recognize the six-vertex model configurationsunder the form of north-east going paths, in which the weight b = 0 (red paths cannot straightto the right). Taking into accounts the weights, this is exactly the free fermionic five-vertex modelconsidered at the end of section 2.4.3 (Fig. 13). Furthermore, if we continue the red lines to theleft so they all end on the same horizontal line, we find exactly the configurations that contributeto the Schur function s λ , where λ is encoded by the sequence of red and green endpoints at the top(which one can extend into an infinite sequence by filling with red dots at the left and green dotsat the right). Finally, by comparing the weights, we find the equality:Ξ σ ( x , . . . , x n | , . . . , 0) = s λ ( x , . . . , x n )In this case, the Ξ σ ( x , . . . , x n | y , . . . , y n ) are usually called factorial Schur functions. They areessentially the same as double Schur functions, see [81, 80] for a detailed discussion.5.3. Orbital varieties. We shall move on to more sophisticated objects. In general, orbital vari-eties are the irreducible components of the intersection of a nilpotent orbit (by conjugation) witha Borel sub-algebra inside a Lie algebra. We cannot possibly reproduce the general theory of or-bital varieties, and here, we shall restrict ourselves to a very special type of orbital varieties whichcorresponds to the Temperley–Lieb algebra (for more general orbital varieties from an “integrable”point of view, see [24, 25]).5.3.1. Geometric description. We use the same notations as in section 5.2; but here all matricesare of even size N = 2 n . Furthermore, call n + = { strict upper triangular matrices } ⊂ b + .Consider the affine scheme O N = { M ∈ n + : M = 0 } hat is upper triangular matrices that square to zero. We have the following description of itsirreducible components: they are indexed by link patterns π ∈ P n . To each π we associate theupper triangular matrix π < with entries ( π < ) ij equal to 1 if i < j and ( i, j ) paired by π , 0 otherwise.For example, π = π < = ... ... Then the corresponding irreducible component O π is given by the equations(5.7) O π = { M ∈ g : M = 0 and rank M i j ≤ rank π
In general, Joseph polynomials are known to be relatedto representations of the corresponding Weyl group, in our case the symmetric group. Note thatthe latter is nothing but the limit q → ± q KZ equation.Consider the Temperley–Lieb algebra with parameter τ = 2 (or q = − 1) acting as in section3.2.5 on the space of link patterns. The rational R -matrix is of the formˇ R i ( u ) = ( a − u ) + u e i a + u where we recall that graphically, e i = , and the identity is 1 = . It can be deducedfrom the trigonometric R -matrix (4.1) by sending q → − q = − e − ~ a / , z = e ~ u, ~ → rational q KZ system isˇ R i ( x i − x i +1 )Ω N ( x , . . . , x N ) = Ω( x , . . . , x i +1 , x i , . . . , x N )(5.11) ρ − Ω N ( x , . . . , x N ) = ( − n − Ω N ( x , . . . , x N , x + 3 a )(5.12)It can be obtained from the (trigonometric) q KZ system (4.2,4.3) by once again sending q to − q = − e − ~ a / , z i = e − ~ x i , ~ → N , the vector of Joseph–Melnikov polynomials defined in the previous section,solves (5.11,5.12), and in fact coincides with the ~ → − n ( n − / ~ − n ( n − Ψ N , where Ψ N is the solution of the trigonometric q KZ system that was discussed in section 4.2.We can proceed analogously to section 4.2.1 and rewrite (5.11) in components by separating itinto two cases: ( i, i + 1) not paired in π : ∂ i (cid:18) Ω π a + x i − x i +1 (cid:19) = 0(5.13) ( i, i + 1) paired in π : − ( a + x i − x i +1 ) ∂ i Ω π = X π ′ = π,e i · π ′ = π Ω π ′ (5.14)We shall now discuss the geometric meaning of (5.13,5.14). Since we know the base case (5.9),this will suffice to prove that Ω N , the vector of multidegrees, satisfies the whole q KZ system.Still, it would be satisfactory to have a geometric interpretation of (5.12) too. Unfortunately, it iscurrently unknown. Note that the effect of the r.h.s. of (5.12) on multidegrees can be quite drastic:for example, starting from the example (5.10), one goes back to the base case 0 , cf (5.9); but sodoing, two quadratic equations have turned into linear equations! Remark: one can write a rational q KZ equation which is analogous to (4.5) but with additivespectral parameters. It should not be confused with the Knizhnik–Zamolodchikov (KZ) equation:the latter is recovered by the further limit a → 0, turning the difference equation into a differentialequation. .3.3. The Hotta construction. The Hotta construction [37] is intended to explain the Joseph rep-resentation (Weyl group representation on Joseph polynomials) for an arbitrary orbital variety.When extended to our scaling action, it will produce the exchange relation (5.11). For more detailssee [25, 59].The idea, as in the Schubert case, is to try to “sweep” with a subgroup L i , here acting byconjugation, and to apply the lemma of section 5.2.4. We need to be a little careful that ourembedding space, n + , is not L i -invariant, so we must translate multidegrees in n + to multidegreesin g . It is easy to see thatmdeg g W = Y i ≥ j ( a + x i − x j ) mdeg n + W W ⊂ n + so that the effect of sweeping with L i amounts to a “gauged” divided difference operator ˜ ∂ i :˜ ∂ i f = 1 a + x i +1 − x i ∂ i (( a + x i +1 − x i ) f ) = ( a + x i − x i +1 ) ∂ i f a + x i − x i +1 Now, letting L i and B + ,i act by conjugation, start with an orbital variety O π which accordingto (5.8), is B + ,i -invariant. If one tries to sweep it with L i , one can in general produce non-uppertriangular matrices. In fact, a small calculation shows that this occurs exactly if M i i +1 = 0. So,we must distinguish two cases: • If ( i, i + 1) are not paired in π , this means that the rank condition at ( i, i + 1) in (5.7) saysthat M i i +1 = 0; which in turn implies that a + x i − x i +1 | Ω π . Furthermore, in this case O π is L i -invariant, so that ˜ ∂ i Ω π = 0. This is exactly (5.13). • If ( i, i + 1) is a pair in π , then generically M i i +1 = 0 in O π . We then proceed in two steps. Cutting : first we intersect O π with the hyperplane M i i +1 = 0. Since the intersection istransverse, the effect on the multidegree is to multiply by the weight of the hyperplane,which is a + x i − x i +1 . Sweeping : now we can sweep with L i and we stay inside O N . Onecan check that the map µ is generically one-to-one, and by dimension argument the imagemust be a union of orbital varieties (plus possibly some lower-dimensional pieces). Theclaim, which we shall not attempt to justify here, is that the orbital varieties thus obtainedare exactly the proper preimages of π under e i . So we find − ˜ ∂ i (( a + x i − x i +1 )Ω π ) = P π ′ = π,e i · π ′ = π Ω π ′ , which is equivalent to (5.14).5.3.4. Recurrence relations and wheel condition. Several other constructions have simple geomet-ric meaning. We mention in passing here the meaning of recurrence relations and of the wheelcondition.Set x i +1 = x i + a . This gives the weight of 0 to M i i +1 . Roughly, this corresponds to looking atwhat happens when M i i +1 → ∞ (this is more or less clear from the integral formula of section 5.1.2;for a more precise statement, see [57]). To remain inside O N , writing the equations ( M ) j i +1 = 0and ( M ) ij = 0, one concludes that one must have M ji = 0 and M i +1 j = 0 for all j . At this stageone notes that the entries M j i +1 and M ij are unconstrained by the equations. Removing the i th and ( i + 1) st rows and columns reduces O N to O N − .What we have just shown is that when M i i +1 → ∞ , being in O N amounts to setting a certainnumber of entries to zeroes and then the result is some irrelevant linear space times O N − . One canbe a bit more careful and do this reasoning at the level of each irreducible component: it is easyto see that only the components that have a pair ( i, i + 1) survive (the others satisfy M i i +1 = 0), nd that O ϕ i π gets sent to O π . Finally, we get the recurrence relations:(5.15) Ω π ( . . . , x i +1 = x i + a , . . . ) = π Im ϕ ii − Y j =1 ( a + x j − x i ) n Y j = i +2 (2 a + x i − x j )Ω π ′ ( x , . . . , x i − , x i +2 , . . . , x N ) π = ϕ i π ′ to be compared with (4.14).Similarly, if one sets x k = x j + 2 a = x i + a , i < j < k , then the equation ( M ) ik = 0 cannotpossibly be satisfied since it contains the infinite term M ij M jk , so all multidegrees must vanish,which is nothing but the wheel condition.5.4. Brauer loop scheme. In this section we introduce a new affine scheme whose existencewas suggested by the underlying integrable model. The subject has an interesting history, whichwe summarize now. In 2003, Knutson, in his study of the commuting variety, introduced theupper-upper scheme [55]; one of its components is closely related to the commuting variety, and inparticular has the same degree. In an a priori unrelated development, de Gier and Nienhuis [18]studied the Brauer loop model, a model of crossing loops which is completely similar to the one thatwe described in section 3.3.1, except crossing plaquettes ( ) are allowed. They found that theentries of the ground state (or steady state of the corresponding Markov process) are again integers,and observed empirically that certain entries coincide with the degrees of the irreducible componentsof the upper-upper scheme. This mysterious connection required an explanation. A partial one wasgiven in [26], where the corresponding inhomogeneous model was introduced and it was suggestedthat this generalization corresponds to going over from degrees to multidegrees. Also, an idea of thegeometric action of the Brauer algebra was given. But some entries remained unidentified. In [58],the Brauer loop scheme was introduced, and it was proved that its top-dimensional componentsproduce all the entries of the ground state. Finally, in the recent preprint [59], it is shown thatmore generally, an appropriate polynomial solution of the q KZ system associated to the Braueralgebra produces multidegrees of these components.5.4.1. Geometric description. There are several equivalent descriptions of the Brauer loop scheme,see [58, 59]. Here we try to present it in a way which best respects the underlying symmetries.Let N be an integer. Consider complex upper triangular matrices that are infinite in bothdirections and that are periodic by shift by ( N, N ): R Z mod N = { M = ( M ij ) i,j ∈ Z : M ij = M i + N j + N ∀ i, j ∈ Z } R Z mod N is an algebra. Let S = ( δ i,j − ) ∈ R Z mod N be the shift operator. Then we define thealgebra ˆ b + to be the quotient ˆ b + = R Z mod N / (cid:10) S N (cid:11) ˆ b + is finite-dimensional: dim ˆ b + = N . Inside ˆ b + we have its radical ˆ n + , which consists of (classesof) matrices with zero diagonal entries, and the group ˆ B + of its invertible elements, i.e. withnon-zero diagonal entries.The Brauer loop scheme is then defined as E N = { M ∈ ˆ n + : M = 0 } For a different point of view on the origin of E N , see section 1.3 of [59]. n all that follows, we take N to be even, N = 2 n . We define a crossing link pattern to be aninvolution of Z /N Z without fixed points. Their set is denoted by P cr n and is of cardinality (2 n − P cr = , , The irreducible components of E N are known to be indexed by crossing link patterns [58, 94]. If π ∈ P cr n , then E π = { M ∈ E N : ( M ) i,i + N = ( M ) j,j + N ⇔ i = j or i = π ( j ) mod N } Note that this definition (i) makes sense because ( M ) i,i + N , despite being undefined for a genericelement of ˆ b + (it is killed by the quotient by S N ), is actually well-defined for elements of ˆ n + and(ii) implies that the N numbers ( M ) i,i + N always come in pairs for M ∈ E N , which is somewhatsurprising (an elementary proof would be nice, as opposed to the proof of [58]).We can also describe E π as the closure of a union of orbits:(5.16) E π = ˆ B + · t π < ˆ B + acts by conjugation, t π < = { M ∈ ˆ b + : M ij = 0 ⇒ i = π ( j ) mod N } Finally, there is a conjectural description in terms of equations:(5.17) E π = { M ∈ ˆ b + : M = 0 , ( M ) i,i + N = ( M ) π ( i ) ,π ( i )+ N , rank M i j ≤ rank π
1. Thus,(5.18) χ n = Υ χ n = N Y i =1 i + n − Y j = i +1 ( a + x i − x j ) At the opposite end, we find the non-crossing link patterns of P n . Among them, we have0 n (or any of its rotations), which has all pairings parallel. In the Brauer loop scheme, theyplay a special role: Knutson proved, in the context of the upper-upper scheme [55], thatthere is a Gr¨obner degeneration of C n × V to E n , where V is some irrelevant vector space,and C n is the commuting variety: C n = { ( X, Y ) ∈ gl ( n, C ) : XY = Y X }• Explicit examples can become quite complicated; even in size N = 4, we find, with thenotation b = a − ǫ : χ = E χ = { M ∈ ˆ n + : M = M = M = M = 0 } Υ χ =( a + x − x )( a + x − x )( a + x − x )( b + x − x )0 = E = M ∈ ˆ n + : M = M = 0 M M + M M = 0 M M + M M = 0 M M − M M = 0 Υ =( a + x − x )( a + x − x )( a + ab + b − b x + a x + x x − a x − x x + b x − x x + x x )1 = ρ (0 ) = E = M ∈ ˆ n + : M = M = 0 M M + M M = 0 M M + M M = 0 M M − M M = 0 Υ =( a + x − x )( b + x − x )( a + 2 ab + b x − a x − x x + a x + x x − b x + x x − x x )The last two varieties are not complete intersections.5.4.2. The Brauer algebra. The Brauer algebra is defined by generators f i , e i , i = 1 , . . . , N − 1, andrelations e i = τ e i e i e i ± e i = e i f i = 1 ( f i f i +1 ) = 1 f i e i = e i f i = e i e i f i f i +1 = e i e i +1 = f i +1 f i e i +1 e i +1 f i f i +1 = e i +1 e i = f i f i +1 e i e i e j = e j e i f i f j = f j f i e i f j = f j e i | i − j | > f i are generatorsof the symmetric group S N , whereas the e i generate a Temperley–Lieb algebra.There is an associated solution of the Yang–Baxter equation, namely(5.20) ˇ R i ( u ) = a ( a − u ) + a u e i + (1 − τ / u ( a − u ) f i ( a + u )( a − (1 − τ / u ) he Brauer algebra acts naturally on crossing link patterns. The graphical rules are the sameas in the previous sections, and we shall not illustrate them again. If π is viewed as an involution,then f i acts by conjugation by the transposition ( i, i + 1), whereas e i creates new cycles ( i, i + 1)and ( π ( i ) , π ( i + 1)) – unless π ( i ) = i + 1, in which case it multiplies the state by τ .We now claim that the vector Υ N of multidegrees Υ π of the irreducible components of the Brauerloop scheme solves the q KZ system associated to the Brauer loop model [59], which we can write(5.21) ˇ R i ( x i − x i +1 )Υ N = s i Υ N ∀ i ∈ Z where s i permutes x i + kN and x i +1+ kN for all k , on condition that the following identification ofthe parameters is made:(5.22) τ = 2( a − ǫ )2 a − ǫ We could also add the cyclicity condition (which is obviously satisfied by Υ N by rotational in-variance), but note that it could at most be marginally stronger than (5.21) (and in fact, it isnot), because we have imposed (5.21) for all integer values of i . In other words, this is already an“affinized” version of the exchange relation because of our shifted periodicity property x i + N = x i + ǫ .For future use, let us write in components (5.21). We find the usual dichotomy: π ( i ) = i + 1 : − ( a + x i − x i +1 )(( a + b ) ∂ i + s i ) (cid:18) Υ π a + x i − x i +1 (cid:19) = Υ f i · π (5.23) π ( i ) = i + 1 : − ( a + x i − x i +1 )( a + b + x i +1 − x i ) ∂ i Υ π = ( a + b ) X π ′ = π,e i · π ′ = π Υ π ′ (5.24)with the convenient notation b = a − ǫ . Remark: The Brauer algebra is the rational limit of the BWM algebra considered in [88].5.4.3. Geometric action of the Brauer algebra. This is the most technical part of [59], which weshall only sketch here. Once again, it follows the same general idea of trying to “sweep” with asubgroup ˆ L i analogous to the L i used so far. In our setting of infinite periodic matrices, this ˆ L i consists of invertible matrices which are the identity everywhere except in the entries M ab with a, b ∈ { i + kN, i + 1 + kN } for some k ; and ˆ B + ,i = ˆ L i ∩ ˆ B + . The additional subtlety comes from thefact that contrary to the case of orbital varieties, the condition M = 0 is not stable by conjugationby ˆ L i (i.e. (cid:10) S N (cid:11) is not stable by conjugation by ˆ L i , since the latter is outside ˆ b + ). We shall haveto reimpose one equation, namely ( M ) i +1 i + N = 0, after sweeping.So, letting ˆ L i and ˆ B + ,i act by conjugation, start with a component E π which according to (5.16),is ˆ B + ,i -invariant. As usual we have to distinguish the two cases: • π ( i ) = i + 1: in this case we can directly sweep with ˆ L i , and then cut with ( M ) i +1 ,i + N . Asshown in [26, 58, 59] with varying levels of rigor and clarity, the result is precisely E π ∪ E f i · π .So we find, at the level of multidegrees, − ( A + B + x i +1 − x i ) ˜ ∂ i Υ π = Υ π + Υ f i · π which reduces after a few manipulations to (5.23). • π ( i ) = i + 1: this time we first cut with M i i +1 = 0, sweep with ˆ L i (throwing away theˆ L i -invariant piece which cannot contribute to the multidegree calculation), and then cut again with ( M ) i +1 i + N . We lost one dimension in the process, so the result cannot simplybe a union of components. In fact, one can show that it is S π ′ = π,e i · π ′ = π E π ∩ { ( M ) i i + N =( M ) i +1 i + N +1 } . This results directly in the multidegree identity (5.24). ecurrence relations can also be written and interpreted geometrically as in section 5.3.4, butwe shall omit them for brevity. Let us simply mention the wheel condition: the Υ π satisfyΥ π ( x i = x, x j = x + a , x k = x + 2 a ) = 0 i < j < k < i + N (cf the appendix B of [88]). This stems geometrically from the equation ( M ) ik = 0 (which is notkilled by the quotient by S N since k < i + N ), which cannot be satisfied when M ij , M jk → ∞ .5.4.4. The degenerate limit. There are two special values of the parameter τ . One, on which weshall not dwell, is τ = 1, which corresponds to ǫ = 0 in (5.22). This is the value for which Υ N becomes the ground state eigenvector of an integrable transfer matrix, that of [18], and is thesubject of [26, 58]. Various interesting results can be obtained, including sum rules. In particular,in the homogeneous limit x i → τ = 2. Indeed, we see that the coefficient of f i inthe R -matrix (5.20) vanishes. In fact we recover this way the Temperley–Lieb solution of rationalYang–Baxter equation, which was used in connection with orbital varieties. What is the geometricmeaning of this reduction?Plugging τ = 2 into (5.22), we note that there is no solution for ǫ , since ǫ → ∞ when τ → ǫ → ∞ in the polynomials Υ π . We consider them as polynomialsin z , . . . , z N , a and ǫ (this choice breaks the rotational invariance z i → z i +1 ) and keep the formerfixed while sending the latter to infinity. Geometrically, the situation is as follows. Let us computethe weights of the entries M ij of a matrix M ∈ ˆ b + in terms of z , . . . , z N , a and b = a − ǫ . Due tothe periodicity, we can always assume i to be between 1 and N , and due to the quotient we shouldthen consider i ≤ j < i + N . The result is that one finds two categories of entries:wt( M ij ) = ( a + z i − z j ≤ i ≤ j ≤ N b + z i − z j − N ≤ j − N < i ≤ N This amounts to subdividing ˆ b + (a vector space of dimension N ) asˆ b + = U L where U is an upper triangular matrix, and L a strict lower triangular matrix. These two pieceshave quite different destinies as τ → 2: the weights of U remain unchanged and are identical tothose that we have used in section 5.3 for orbital varieties, whereas the weights of L diverge. Itturns out that this limit corresponds to killing off this lower triangular part (as is clear from thedefinition of the multidegree as an integral, see section 5.1.2); and after taking the quotient, it iseasy to see that we are left with b + , the algebra of upper triangular matrices. Let us call p thisprojection ( U, L ) U from ˆ b + to b + .At the level of components, here is what happens: if π is a non-crossing link pattern, then p ( E π ) = O π , the corresponding orbital variety. Translated into multidegrees, this means:Υ π ( z , . . . , z N , a , b ) B →∞ ∼ b n ( n − Ω π ( z , . . . , z N , a )In general, crossing link patterns will project to lower dimensional B + -orbit closures [59].Interestingly, the matrix Schubert varieties and double Schubert polynomials of section 5.2 alsoappear as a special case in the τ = 2 limit. Indeed, consider the permutation sector , that is the ubset of P cr n consisting of involutions such that π ( { , . . . , n } ) = { n + 1 , . . . , n } . Such involutionsare in bijection with permutations σ ∈ S n , according to σ ( i ) = π ( n + 1 − i ) − n , i = 1 , . . . , n : σ = ←→ π = flip ←→ π = (the last flip is purely cosmetic and due to the fact that we have written labels of link patternscounterclockwise up to now).In the permutation sector, the structure of E π is as follows: E π = X Y ⋆⋆ If one discards the unconstrained entries (represented by ⋆ ), one recognizes the pairs of n × n matrices ( X, Y ) in terms of which the upper-upper scheme of Knutson is defined [55]. In fact theunion of E π where π runs over the permutation sector is exactly up to these irrelevant entries theupper-upper scheme.Now the projection of such components E π , that is here ( X, Y ) X , is essentially the corre-sponding matrix Schubert variety S σ . There are various ways to see that; one can prove it rigorouslyby using the description in terms of orbits; or one can compare the defining equations (5.1) of ma-trix Schubert varieties to the (conjectured) defining equations (5.17) of E π (which, in the case ofthe permutation sector, reduce to the first conjecture of section 3 of [55]). In any case, the resultis more precisely that there is a vertical flip, so that p ( E π ) ≃ S σ σ . In terms of multidegrees, thismeans thatΥ π ( x , . . . , x n , a , b ) B →∞ ∼ b n ( n − −| σ | Y ≤ i 2. The operators t i then satisfy thenil-Hecke algebra, and one can check that in the permutation sector, where the e i part of the R -matrix never contributes when i = n , the exchange relation (5.21) reduces to (5.3) if i < n or to(5.4) if i > n . Acknowledgements I would like to thank A. Knutson, J.-M. Maillet, F. Smirnov who accepted to be members ofmy jury, and especially V. Pasquier, N. Reshetikhin and X. Viennot who agreed to referee thismanuscript. I would also like to specially thank J.-B. Zuber, who besides collaborating with me onnumerous occasions, has regularly advised in my work and provided guidance during the preparationof this habilitation. would like to thank my colleagues at my former laboratory (LPTMS Orsay) and new one(LPTHE Jussieu), especially their directors S. Ouvry and O. Babelon who welcomed me in theirlabs and were always open for discussions.My thanks to all my collaborators, here listed in the order of the number of co-publications (andalphabetically in case of ties): P. Di Francesco, with whom I have had the pleasure to have a mostfruitful collaboration, which is the basis for a good part of the results in the manuscript; J. Jacobsen,J.-B. Zuber; V. Korepin, N. Andrei, J. de Gier, T. Fonseca, V. Kazakov, A. Knutson (the onlyperson with whom I wrote a paper without ever meeting in person), P. Pyatov, A. Razumov,G. Schaeffer, Yu. Stroganov. My interaction with all of them has been influential in my scientificcareer.Finally, I would like to thank C. Le Vaou whose invaluable help with the administrative issuesduring all these years at LPTMS made it possible for me to concentrate on my research; and myfamily and friends for their support during the writing process.My work was supported by European Union Marie Curie Research Training Networks “EN-RAGE” MRTN-CT-2004-005616, “ENIGMA” MRT-CT-2004-5652, European Science Foundationprogram “MISGAM” and Agence Nationale de la Recherche program “GIMP” ANR-05-BLAN-0029-01. References [1] D. Allison and N. Reshetikhin, Numerical study of the 6-vertex model with domain wall boundary conditions ,Ann. Inst. Fourier (Grenoble) (2005), no. 6, 1847–1869, arXiv:cond-mat/0502314 . MR MR2187938[2] G. Andrews, Plane partitions. V. The TSSCPP conjecture , J. Combin. Theory Ser. A (1994), no. 1, 28–39.MR MR1273289[3] M. Batchelor, J. de Gier, and B. Nienhuis, The quantum symmetric XXZ chain at ∆ = − / , alternating-sign matrices and plane partitions , J. Phys. A (2001), no. 19, L265–L270, arXiv:cond-mat/0101385 .MR MR1836155[4] R. Baxter, Exactly solved models in statistical mechanics , Academic Press, 1982.[5] A. Belavin, A. Polyakov, and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum fieldtheory , Nucl. Phys. B (1984), no. 2, 333, doi .[6] P. Bleher and V. Fokin, Exact solution of the six-vertex model with domain wall boundary conditions. Disorderedphase , Comm. Math. Phys. (2006), no. 1, 223–284, arXiv:math-ph/0510033 . MR MR2249800[7] P. Bleher and K. Liechty, Exact solution of the six-vertex model with domain wall boundary conditions. Criticalline between ferroelectric and disordered phases , J. Stat. Phys. (2009), no. 3, 463–485, arXiv:0802.0690 .MR MR2485725[8] , Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase ,Comm. Math. Phys. (2009), no. 2, 777–801, arXiv:0712.4091 . MR MR2472044[9] D. Bressoud, Proofs and confirmations: The story of the alternating sign matrix conjecture , MAA Spectrum,Mathematical Association of America, Washington, DC, 1999. MR MR1718370[10] Harish Chandra, Differential operators on a semi-simple Lie algebra , Amer. J. Math. (1957), 87–120.[11] I. Cherednik, Quantum Knizhnik–Zamolodchikov equations and affine root systems , Comm. Math. Phys. (1992), no. 1, 109–136, http://projecteuclid.org/getRecord?id=euclid.cmp/1104251785 . MR MR1188499[12] , Double affine Hecke algebras , London Mathematical Society Lecture Note Series, vol. 319, CambridgeUniversity Press, Cambridge, 2005. MR MR2133033[13] H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition , New York J. Math. (1998),137–165, arXiv:math/9801059 . MR MR1641839[14] F. Colomo and A. Pronko, The arctic circle revisited , 2007, arXiv:0704.0362 .[15] , Emptiness formation probability in the domain-wall six-vertex model , Nuclear Phys. B (2008),no. 3, 340–362, arXiv:0712.1524 . MR MR2411855[16] , The limit shape of large alternating sign matrices , 2008, arXiv:0803.2697 . 17] J. de Gier, Loops, matchings and alternating-sign matrices , Discrete Math. (2005), no. 1-3, 365–388, arXiv:math/0211285 . MR MR2163456[18] J. de Gier and B. Nienhuis, Brauer loops and the commuting variety , J. Stat. Mech. Theory Exp. (2005), no. 1,006, 10 pp, arXiv:math.AG/0410392 . MR MR2114232[19] J. de Gier and P. Pyatov, Factorised solutions of Temperley–Lieb q KZ equations on a segment , arXiv:0710.5362 .[20] J. de Gier, P. Pyatov, and P. Zinn-Justin, Punctured plane partitions and the q -deformed Knizhnik–Zamolodchikov and Hirota equations , J. Combin. Theory Ser. A (2009), 772–794, arXiv:0712.3584 , doi .[21] D. Dei Cont and B. Nienhuis, The packing of two species of polygons on the square lattice , J. Phys. A (2004),3085–3100, arXiv:cond-mat/0311244 .[22] P. Di Francesco, Totally symmetric self-complementary plane partitions and the quantum Knizhnik–Zamolodchikov equation: a conjecture , J. Stat. Mech. Theory Exp. (2006), no. 9, P09008, 14 pp, arXiv:cond-mat/0607499 . MR MR2278472[23] P. Di Francesco and P. Zinn-Justin, Around the Razumov–Stroganov conjecture: proof of a multi-parameter sumrule , Electron. J. Combin. (2005), Research Paper 6, 27 pp, arXiv:math-ph/0410061 . MR MR2134169[24] , Quantum Knizhnik–Zamolodchikov equation, generalized Razumov–Stroganov sum rules and extendedJoseph polynomials , J. Phys. A (2005), no. 48, L815–L822, arXiv:math-ph/0508059 , doi . MR MR2185933[25] , From orbital varieties to alternating sign matrices , 2006, extended abstract for FPSAC’06, arXiv:math-ph/0512047 .[26] , Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties , Comm. Math.Phys. (2006), no. 2, 459–487, arXiv:math-ph/0412031 . MR MR2200268[27] , Quantum Knizhnik–Zamolodchikov equation: reflecting boundary conditions and combinatorics , J. Stat.Mech. Theory Exp. (2007), no. 12, P12009, 30 pp, arXiv:0709.3410 , doi . MR MR2367185[28] , Quantum Knizhnik–Zamolodchikov equation, totally symmetric self-complementary plane partitions andalternating sign matrices , Theor. Math. Phys. (2008), no. 3, 331–348, arXiv:math-ph/0703015 , doi .[29] P. Di Francesco, P. Zinn-Justin, and J.-B. Zuber, A bijection between classes of fully packed loops and plane par-titions , Electron. J. Combin. (2004), no. 1, Research Paper 64, 11 pp, arXiv:math/0311220 . MR MR2097330[30] , Determinant formulae for some tiling problems and application to fully packed loops , Ann. Inst. Fourier(Grenoble) (2005), no. 6, 2025–2050, arXiv:math-ph/0410002 . MR MR2187944[31] L. D. Faddeev, How Algebraic Bethe Ansatz works for integrable model , 1996, arXiv:hep-th/9605187 .[32] S. Fomin and A. Kirillov, The Yang–Baxter equation, symmetric functions, and Schubert polynomials , Proceed-ings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), vol. 153,1996, pp. 123–143. MR MR1394950[33] S. Fomin and R. Stanley, Schubert polynomials and the nil-Coxeter algebra , Adv. Math. (1994), no. 2,196–207. MR MR1265793[34] T. Fonseca and P. Zinn-Justin, On the doubly refined enumeration of alternating sign matrices and totallysymmetric self-complementary plane partitions , Electron. J. Combin. (2008), Research Paper 81, 35 pp, arXiv:0803.1595 . MR MR2411458[35] I. Frenkel and N. Reshetikhin, Quantum affine algebras and holonomic difference equations , Commun. Math.Phys. (1992), 1–60, http://projecteuclid.org/euclid.cmp/1104249974 .[36] I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae , Adv. in Math. (1985),no. 3, 300–321. MR MR815360[37] R. Hotta, On Joseph’s construction of Weyl group representations , Tohoku Math. J. (2) (1984), no. 1, 49–74.MR MR733619[38] C. Itzykson and J.-B. Zuber, The planar approximation. II , J. Math. Phys. (1980), no. 3, 411–421.MR MR562985[39] A. Izergin, D. Coker, and V. Korepin, Determinant formula for the six-vertex model , J. Phys. A (1992),no. 16, 4315–4334. MR MR1181591[40] A. G. Izergin, Partition function of a six-vertex model in a finite volume , Dokl. Akad. Nauk SSSR (1987),no. 2, 331–333, http://adsabs.harvard.edu/abs/1987SPhD...32..878I . MR MR919260[41] J. Jacobsen and J. Kondev, Field theory of compact polymers on the square lattice , Nucl. Phys. B (1998),635–688, arXiv:cond-mat/9804048 .[42] , Conformal field theory of the Flory model of polymer melting , Phys. Rev. E (2004), no. 6, 066108, arXiv:cond-mat/0209247 , doi .[43] J. Jacobsen and P. Zinn-Justin, Algebraic Bethe Ansatz for the FPL model , J. Phys. A (2004), no. 29,7213–7225, arXiv:math-ph/0402008 , doi . MR MR2078954 44] M. Jimbo and T. Miwa, Solitons and infinite-dimensional Lie algebras , Publ. Res. Inst. Math. Sci. (1983),no. 3, 943–1001, http://projecteuclid.org/euclid.prims/1195182017 . MR MR723457[45] , Algebraic analysis of solvable lattice models , CBMS Regional Conference Series in Mathematics, vol. 85,Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1995. MR MR1308712[46] W. Jockush, J. Propp, and P. Shor, Random domino tilings and the arctic circle theorem , arXiv:math.CO/9801068 .[47] K. Johansson, The arctic circle boundary and the Airy process , Ann. Probab. (2005), no. 1, 1–30, arXiv:math/0306216 . MR MR2118857[48] A. Joseph, Orbital varieties, Goldie rank polynomials and unitary highest weight modules , Algebraic and analyticmethods in representation theory (Sønderborg, 1994), Perspect. Math., vol. 17, Academic Press, San Diego,CA, 1997, pp. 53–98. MR MR1415842[49] M. Kasatani, Subrepresentations in the polynomial representation of the double affine Hecke algebra of type GL n at t k +1 q r − = 1, Int. Math. Res. Not. (2005), no. 28, 1717–1742, arXiv:math/0501272 . MR MR2172339[50] M. Kasatani and Y. Takeyama, The quantum Knizhnik–Zamolodchikov equation and non-symmetric Macdonaldpolynomials , Funkcial. Ekvac. (2007), no. 3, 491–509, arXiv:math/0608773 . MR MR2381328[51] P. Kasteleyn, Graph theory and crystal physics , Graph Theory and Theoretical Physics, Academic Press, Lon-don, 1967, pp. 43–110. MR MR0253689[52] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. (1979),no. 2, 165–184, doi . MR MR560412[53] R. Kenyon, Lectures on dimers , .[54] A. Kirillov, Jr. and A. Lascoux, Factorization of Kazhdan–Lusztig elements for Grassmannians , Combinatorialmethods in representation theory (Kyoto, 1998), Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000,pp. 143–154, arXiv:math.CO/9902072 . MR MR1864480[55] A. Knutson, Some schemes related to the commuting variety , J. Algebraic Geom. (2005), no. 2, 283–294, arXiv:math.AG/0306275 . MR MR2123231[56] A. Knutson and E. Miller, Gr¨obner geometry of Schubert polynomials , Ann. of Math. (2) (2005), no. 3,1245–1318. MR MR2180402[57] A. Knutson, E. Miller, and A. Yong, Gr¨obner geometry of vertex decompositions and of flagged tableaux , Journalf¨ur die reine und angewandte Mathematik (2007), arXiv:math.CO/0502144 .[58] A. Knutson and P. Zinn-Justin, A scheme related to the Brauer loop model , Adv. Math. (2007), no. 1,40–77, arXiv:math.AG/0503224 . MR MR2348022[59] , The Brauer loop scheme and orbital varieties , 2009, preprint.[60] V. Korepin, Calculation of norms of Bethe wave functions , Comm. Math. Phys. (1982), no. 3, 391–418.MR MR677006[61] V. Korepin and P. Zinn-Justin, Thermodynamic limit of the six-vertex model with domain wall boundary con-ditions , J. Phys. A (2000), no. 40, 7053–7066, arXiv:cond-mat/0004250 , doi . MR MR1792450[62] G. Kuperberg, Another proof of the alternating-sign matrix conjecture , Internat. Math. Res. Notices (1996),no. 3, 139–150, arXiv:math/9712207 . MR MR1383754[63] A. Lascoux, Symmetric functions and combinatorial operators on polynomials , CBMS Regional ConferenceSeries in Mathematics, vol. 99, Published for the Conference Board of the Mathematical Sciences, Washington,DC, 2003. MR MR2017492[64] A. Lascoux and M.-P. Sch¨utzenberger, Polynˆomes de Kazhdan & Lusztig pour les grassmanniennes , Youngtableaux and Schur functors in algebra and geometry (Toru´n, 1980), Ast´erisque, vol. 87, Soc. Math. France,Paris, 1981, pp. 249–266. MR MR646823[65] G. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents. I. Half-plane exponents ,Acta Math. (2001), no. 2, 237–273. MR MR1879850[66] , Values of Brownian intersection exponents. II. Plane exponents , Acta Math. (2001), no. 2, 275–308.MR MR1879851[67] , Values of Brownian intersection exponents. III. Two-sided exponents , Ann. Inst. H. Poincar´e Probab.Statist. (2002), no. 1, 109–123. MR MR1899232[68] E. Lieb, Exact solution of the F model of an antiferroelectric , Phys. Rev. Lett. (1967), no. 24, 1046–1048, doi .[69] , Exact solution of the problem of the entropy of two-dimensional ice , Phys. Rev. Lett. (1967), no. 17,692–694, doi .[70] , Exact solution of the two-dimensional Slater KDP model of a ferroelectric , Phys. Rev. Lett. (1967),no. 3, 108–110, doi .[71] , Residual entropy of square ice , Phys. Rev. (1967), no. 1, 162–172, doi . 72] B. Lindstr¨om, On the vector representations of induced matroids , Bull. London Math. Soc. (1973), 85–90.MR MR0335313[73] I. Macdonald, Affine Hecke algebras and orthogonal polynomials , Ast´erisque (1996), no. 237, Exp. No. 797, 4,189–207, S´eminaire Bourbaki, Vol. 1994/95. MR MR1423624[74] P. MacMahon, Combinatory analysis , Cambridge University Press, 1915.[75] A. Melnikov, B -orbits in solutions to the equation X = 0 in triangular matrices , J. Algebra (2000), no. 1,101–108. MR MR1738254[76] E. Miller and B. Sturmfels, Combinatorial commutative algebra , Graduate Texts in Mathematics, vol. 227,Springer-Verlag, New York, 2005. MR MR2110098[77] W. Mills, D. Robbins, and H. Rumsey, Jr., Alternating sign matrices and descending plane partitions , J. Combin.Theory Ser. A (1983), no. 3, 340–359. MR MR700040[78] W. Mills, D. Robbins, and H. Rumsey, Jr, Self-complementary totally symmetric plane partitions , J. Combin.Theory Ser. A (1986), no. 2, 277–292. MR MR847558[79] T. Miwa, On Hirota’s difference equations , Proc. Japan Acad. Ser. A Math. Sci. (1982), no. 1, 9–12, http://projecteuclid.org/euclid.pja/1195516178 . MR MR649054[80] A. Molev, Comultiplication rules for the double Schur functions and Cauchy identities , 2008, arXiv:0807.2127 .[81] A. Molev and B. Sagan, A Littlewood–Richardson rule for factorial Schur functions , Transactions of the Amer-ican Mathematical Society (1999), no. 11, 4429–4443, arXiv:q-alg/9707028 .[82] B. Nienhuis, Phase transitions and critical phenomena , vol. 11, Academic Press, 1987, Eds. C. Domb and J.Lebowitz.[83] J.D. Noh and D. Kim, Interacting domain walls and the five-vertex model , arXiv:cond-mat/9312001 .[84] I. Nolden, The asymmetric six-vertex model , J. Stat. Phys. (1992), 155–201, doi .[85] S. Okada, Enumeration of symmetry classes of alternating sign matrices and characters of classical groups , J.Algebraic Combin. (2006), no. 1, 43–69, arXiv:math/0408234 , doi . MR MR2218849[86] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of arandom 3-dimensional Young diagram , J. Amer. Math. Soc. (2003), no. 3, 581–603, arXiv:math.CO/0107056 .MR MR1969205[87] V. Pasquier, Scattering matrices and affine Hecke algebras , Low-dimensional models in statistical physics andquantum field theory (Schladming, 1995), Lecture Notes in Phys., vol. 469, Springer, Berlin, 1996, pp. 145–163, arXiv:q-alg/9508002 . MR MR1477945[88] , Incompressible representations of the Birman–Wenzl–Murakami algebra , Ann. Henri Poincar´e (2006),no. 4, 603–619, arXiv:math/0507364 . MR MR2232366[89] , Quantum incompressibility and Razumov Stroganov type conjectures , Ann. Henri Poincar´e (2006),no. 3, 397–421, arXiv:cond-mat/0506075 . MR MR2226742[90] A. Razumov and Yu. Stroganov, Combinatorial nature of the ground-state vector of the O (1) loop model , Teoret.Mat. Fiz. (2004), no. 3, 395–400, arXiv:math/0104216 , doi . MR MR2077318[91] , Bethe roots and refined enumeration of alternating-sign matrices , J. Stat. Mech. Theory Exp. (2006),no. 7, P07004, 12 pp, arXiv:math-ph/0605004 . MR MR2244324[92] A. Razumov, Yu. Stroganov, and P. Zinn-Justin, Polynomial solutions of q KZ equation and ground state of XXZ spin chain at ∆ = − / 2, J. Phys. A (2007), no. 39, 11827–11847, arXiv:0704.3542 , doi . MR MR2374053[93] N. Reshetikhin, Lectures on the six-vertex model , Les Houches lecture notes.[94] B. Rothbach, Equidimensionality of the Brauer loop scheme , 2009, preprint.[95] J. Shore and D. J. Bukman, Coexistence point in the six-vertex model and the crystal shape of fcc materials ,Phys. Rev. Lett. (1994), no. 5, 604–607, doi .[96] F. Smirnov, A general formula for soliton form factors in the quantum sine–Gordon model , J. Phys. A (1986), L575–L578.[97] R. Stanley, A baker’s dozen of conjectures concerning plane partitions , Combinatoire ´enum´erative (Montreal,Que., 1985/Quebec, Que., 1985), Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 285–293.MR MR927770[98] Yu. Stroganov, Izergin–Korepin determinant at a third root of unity , Teoret. Mat. Fiz. (2006), no. 1, 65–76, arXiv:math-ph/0204042 . MR MR2243403[99] B. Sutherland, Exact solution of a two-dimensional model for hydrogen-bonded crystals , Phys. Rev. Lett. (1967), no. 3, 103–104, doi .[100] B. Wieland, A large dihedral symmetry of the set of alternating sign matrices , Electron. J. Combin. (2000),Research Paper 37, 13 pp, arXiv:math/0006234 . MR MR1773294[101] F. Wu, Remarks on the modified potassium dihydrogen phosphate model of a ferroelectric , Phys. Rev. (1968),no. 2, 539–543, doi . Proof of the alternating sign matrix conjecture , Electron. J. Combin. (1996), no. 2, ResearchPaper 13, 84 pp, The Foata Festschrift, arXiv:math/9407211 . MR MR1392498[103] , Proof of a conjecture of Philippe Di Francesco and Paul Zinn-Justin relatedto the q KZ equations and to Dave Robbins’ two favorite combinatorial objects , 2007, .[104] P. Zinn-Justin, The influence of boundary conditions in the six-vertex model , arXiv:cond-mat/0205192 .[105] , Six-vertex model with domain wall boundary conditions and one-matrix model , Phys. Rev. E (2000),no. 3, part A, 3411–3418, arXiv:math-ph/0005008 . MR MR1788950[106] , HCIZ integral and 2D Toda lattice hierarchy , Nuclear Phys. B (2002), no. 3, 417–432, arXiv:math-ph/0202045 . MR MR1912027[107] , Proof of the Razumov–Stroganov conjecture for some infinite families of link patterns , Electron. J.Combin. (2006), no. 1, Research Paper 110, 15 pp, arXiv:math/0607183 . MR MR2274325[108] , Combinatorial point for fused loop models , Comm. Math. Phys. (2007), no. 3, 661–682, arXiv:math-ph/0603018 , doi . MR MR2304471[109] , Littlewood–Richardson coefficients and integrable tilings , Electron. J. Combin. (2009), ResearchPaper 12, arXiv:0809.2392 . Paul Zinn-Justin, LPTMS (CNRS, UMR 8626), Univ Paris-Sud, 91405 Orsay Cedex, France; andLPTHE (CNRS, UMR 7589), Univ Pierre et Marie Curie-Paris6, 75252 Paris Cedex, France. E-mail address : pzinn @ lpthe.jussieu.frpzinn @ lpthe.jussieu.fr