Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
aa r X i v : . [ m a t h - ph ] N ov Quantum Knizhnik–Zamolodchikov Equation:Reflecting boundary conditions and Combinatorics
P. Di Francesco and P. Zinn-JustinWe consider the level 1 solution of quantum Knizhnik–Zamolodchikov equation with re-flecting boundary conditions which is relevant to the Temperley–Lieb model of loops ona strip. By use of integral formulae we prove conjectures relating it to the weighted enu-meration of Cyclically Symmetric Transpose Complement Plane Partitions and relatedcombinatorial objects.09/2007 . Introduction
Since the papers [1,2], there has been a great deal of work on the combinatorialinterpretation of quantum integrable models at special points of their parameter space.The original observation is that the numbers of Alternating Sign Matrices (ASM) and PlanePartitions (PP) in various symmetry classes appear naturally in the ground state entriesof the Temperley–Lieb O ( τ = 1) model of (non-crossings) loops with various boundaryconditions (and related models). The appearance of ASM numbers was developed furtherand to some extent explained by the Razumov–Stroganov conjecture [3] and variants [4,5]interpreting each ground state entry as a number of certain subsets of ASM. The roleof plane partitions remained more obscure until the recent work [6,7] which showed thatthe enumeration of symmetry classes of PP also occurs naturally on condition that oneconsider a slightly more general problem, namely the quantum Knizhnik–Zamolodchikovequation ( q KZ), first introduced in this context in [8], and in which the parameter τ isnow free. This provided a (conjectural) bridge between enumerations of symmetry classesof ASM and PP, which is a fascinating topic of enumerative combinatorics in itself.The present work is concerned more specifically with the case of the Temperley–Lieb loop model (and its q KZ generalization) defined on a strip with reflecting boundaryconditions (the case of periodic boundary conditions was treated similarly in [9]). Thecorresponding ASM were discovered in [4,10]: they are Vertically Symmetric AlternatingSign Matrices (VSASM) of size (2 n + 1) × (2 n + 1) in even strip size N = 2 n , and modifiedVSASM of size (2 n + 1) × (2 n + 3) in odd strip size N = 2 n + 1. As to the PP, they werediscussed in [7]: they are Cyclically Symmetric Transpose Complement Plane Partitions(CSTCPP) [11] in odd strip size, and certain modified CSTCPP (referred to as CSTCPP △ in the following) in even strip size. The conjectures of [7] concerning the τ -enumerationof these plane partitions are the main subject of this work. In Sect. 2 we shall review thebasics of integrable loop models based on the Temperley–Lieb algebra; in Sect. 3 we shalldiscuss the related q KZ equation, and review the conjectures of [7]; in Sect. 4 we introducethe main technical tool, that is certain explicit integrals solving q KZ; and finally in Sect. 5and 6 we prove the conjectures of [7], considering separately even and odd cases.1 . Loop model with reflecting boundary conditions and link patterns
We consider the version with reflecting boundaries of the inhomogeneous O (1) non-crossing loop model [12]. The model is defined on a semi-infinite strip of width N (evenor odd) of square lattice, with centers of the lower edges labelled 1 , , ..., N . On each faceof this domain of the square lattice, we draw at random, say with respective probabilities1 − t i , t i in the i -th column (at the vertical of the point labelled i ) one of the two followingconfigurations (2 . N1 2 . . . (2 . , , ..., N according to their connection via the paths (except for one point if N is odd which isconnected to the infinity along the strip). Such a pattern of connections is called a linkpattern , and an individual pairing is called an arch. The set of link patterns on N pointsis denoted by LP N , and has cardinality (2 n )! / ( n !( n + 1)!) for N = 2 n or N = 2 n − N = 2 n − n but with the point 2 n sent to infinity on the strip: this provides a natural bijectionbetween LP n − and LP n . A link pattern π ∈ LP N may also be viewed as a permutation These boundary conditions are sometimes called “open”, in reference to the equivalent openXXZ spin chain, or “closed”, due to the way the loops close at the boundaries of the strip. Fig. 1:
A sample configuration of the Dense Loop model on a strip of width N = 6 (left). We have indicated the corresponding open non-crossing linkpattern of connection of the points 1 , , , , , π ∈ S N with only cycles of length 2 (except one cycle of length 1 for N odd), and weshall use the notation π ( i ) = j to express that points i and j are connected by an arch.For a pair ( i, j ) such that j = π ( i ) and i < j , we will call i the opening and j the closingof the arch connecting i and j . An example of loop configuration together with its linkpattern are depicted in Fig. 1. We use a standard pictorial representation for link patternsin the form of configurations of non-intersecting arches connecting regularly spaced pointson a line, within the upper-half plane it defines. For odd N , the unmatched point maybe represented as connected to infinity in the upper-half plane via a vertical half-line.We moreover attach a weight τ = − ( q + q − ) = 1 to each loop (hence the denominationO( τ = 1) model, q = − e iπ/ ). We may then compute the probability Prob( π ) for a givenrandomly generated configuration of the loop model on the strip to be connecting theboundary points according to a given link pattern π .In Ref. [12], the model was solved by means of a transfer matrix technique, usingsolutions of the boundary Yang–Baxter equation [13] [14] that parametrize the inhomoge-neous probabilities t i via integrable Boltzmann weights, coded by a standard trigonometric R -matrix. Using the integrability of the system, and following the philosophy of [15], thesuitably renormalized vector of probabilities Ψ ≡ { Ψ π } π ∈ LP N was shown to satisfy thequantum Knizhnik–Zamolodchikov equation with reflecting boundaries for q = − e iπ/ , inthe link pattern basis.In the following, we will consider the more general case of generic q , τ , which doesnot have stricto sensu an interpretation in terms of lattice loop model [16].3 .2. R matrix The R -matrix of the model is an operator acting on link patterns of LP N :ˇ R i,i +1 ( z, w ) = qz − q − wqw − q − z + z − wqw − q − z = qz − q − wqw − q − z I + z − wqw − q − z e i (2 . z and w are complex numbers attached to the points i and i + 1 and where e i , i = 1 , ..., N − T L N ( τ ), subject to therelations e i = τ e i , [ e i , e j ] = 0 if | i − j | > , e i e i ± e i = e i (2 . τ = − q − q − (2 . i+1 = i i+1 kj j ki i+1 = τ i i+1 i Fig. 2:
Action of the Temperley-Lieb generators e i on link patterns.In (2.3), we have depicted the Temperley-Lieb generators as tilted squares with edgecenters connected by pairs. The corresponding action on link patterns should be under-stood as follows (see Fig. 2): assume the points i, i + 1 are connected to say the points j, k in a link pattern π . Then, unless j = i + 1 and k = i , the link pattern π ′ = e i π is identicalto π except that i is now connected to i + 1, and j to k . If j = i + 1 and k = i , the points i, i + 1 are connected to each other in π , and e i π = τ π . The latter is a direct consequence4f the projector condition e i = τ e i , as any link pattern with i connected to i + 1 lies inthe image of e i .The above R -matrix satisfies the Yang–Baxter equation and the unitarity relationˇ R i,i +1 ( z, w ) ˇ R i +1 ,i +2 ( z, x ) ˇ R i,i +1 ( w, x ) = ˇ R i +1 ,i +2 ( w, x ) ˇ R i,i +1 ( z, x ) ˇ R i +1 ,i +2 ( z, w )ˇ R i,i +1 ( z, w ) ˇ R i,i +1 ( w, z ) = I (2 . (c)(b)(a) Fig. 3:
Dyck path (b) associated to a link pattern (a). The former is obtainedas the discrete path on the non-negative integer line: 0 , , , , , , , , , , c ) the box decomposition of the Dyck path.Before turning to the q KZ equation, we wish to emphasize a number of useful prop-erties satisfied by the link patterns, and the action of the Temperley-Lieb generators. Analternative pictorial representation of link patterns is via Dyck paths, namely paths fromand to the origin on the non-negative integer line, with steps of ± N = 10. For even N = 2 n ,we construct the Dyck path by visiting the points of the link pattern from 1 to N , startingfrom the origin of the non-negative integer half-line, and with the following rule: if anarch opens (resp. closes) at i , then the path goes up (resp. down) one step between time i − i . The path is guaranteed to come back to the origin at time 2 n as there are asmany openings as closings of arches, and moreover stays in the non-negative half-line asall arches must first open before closing. In the case of odd N = 2 n −
1, one arch exactly5as an opening and no closing point (it is connected to infinity), hence the path ends up atthe point 1 on the integer half-line. It can be completed into a path of length 2 n by a finalstep to the origin, thus expressing on Dyck paths the abovementioned bijection between LP n − and LP n . The Dyck path is represented in the plane as the (broken-line) graphof the function ( t, h ( t )), t = 0 , , ..., N .Dyck paths allow to endow the set of link patterns with a natural “containment” order,namely π < π ′ iff the Dyck path of π contains strictly that of π ′ . This notion is madeexplicit by introducing the “box decomposition” of any given Dyck path (see Fig. 3 (c)),namely the decomposition of the region of the plane delimited by the path and a brokenline (0 , → (1 , → (2 , → · · · → ( N − , → ( N,
0) if N = 2 n , without the last stepfor N = 2 n −
1, by use of squares of edge √ ◦ . Then a Dyck path δ containsstrictly another δ ′ iff δ is obtained from δ ′ by addition of at least one box. A box addition atposition m consists simply in replacing a portion of path ( m − , h ) → ( m, h − → ( m +1 , h )that visits successively the points h, h − , h of the integer half-line at times m − , m, m + 1,by the portion ( m − , h ) → ( m, h + 1) → ( m + 1 , h ), thus adding a box with center atcoordinates ( m, h ). This may also be described as transforming a local minimum into alocal maximum at position m on the path. The “smallest” link pattern (whose Dyck pathcontains all others) is the pattern π with links π ( i ) = 2 n + 1 − i , i = 1 , , ..., n for N = 2 n ,and i = 2 , ..., n , for N = 2 n −
1, while π (1) = 1. It corresponds to the farthest excursion,reaching point n on the integer half-line. The “largest” link pattern (whose Dyck pathis contained in all others) π max has π max ( i ) = i + 1 for i = 1 , , .., n − N = 2 n and i = 1 , , ..., n − N = 2 n −
1, while π max (2 n −
1) = 2 n −
1. It correspondsto the shortest range excursion, alternating betwen the origin and point 1 on the integerhalf-line. So we have π < π < π max for all π ∈ LP N such that π = π , π max . Finally, weshall denote by β ( π ) the total number of boxes in the box decomposition of the Dyck pathassociated to π . We have for instance β ( π ) = n ( n − / β ( π max ) = 0.The action of e i on link patterns may be easily translated into the language of boxeson Dyck paths. The action of e i may indeed be viewed as a box addition at position i onthe corresponding Dyck paths. Then 3 situations may occur (Fig. 4):(i) The path has a minimum at point i : the box addition transforms it into a maximum.(ii) The path has a maximum at point i : the box-added path is unchanged, but picks upa factor of τ .(iii) The path has a slope at i , namely a succession of two up or two down steps: the boxaddition actually destroys the two rows of boxes at its height and immediately below6 iii)(i)(ii) τ Fig. 4:
Box addition at position i on Dyck paths corresponding to the actionof e i . We have depicted three generic situations for the addition: (i) at a localminimum (ii) at a local maximum (iii) at a slope. Both (ii) and (iii) lead toa Dyck path contained by the original one, while (i) produces a Dyck pathcontaining it, with exactly one additional box.until the other side of the path is reached. The net result may be interpreted as anavalanche, in which the mountain shape between the point of impact and the otherside falls down by two units.This allows to see that among all possible actions of e i on a link pattern π , only oneleads to a “larger” Dyck path (containing π ): e i π = π ′ < π , namely in the situation ( i ),while all other situations lead to π < π ′ = e i π . This observation will be used below.The interpretation of the action of e i on Dyck paths was used in [5] to rephrase thehomogeneous loop model as the stochastic model of a growing interface.
3. The q KZ equation for reflecting boundary condition
The reflecting boundary q KZ equation consists of the following system of equationsfor a vector Ψ which depends polynomially on the variables z , . . . , z N (and q, q − ):ˇ R i ( z i +1 , z i )Ψ N ( z , . . . , z i , z i +1 , . . . , z N ) = Ψ N ( z , . . . , z i +1 , z i , . . . , z N ) (3 . a ) c N ( z N )Ψ N ( z , . . . , z N ) = Ψ N ( z , . . . , z L − , s/z N ) (3 . b ) c ( z )Ψ N ( z , . . . , z N ) = Ψ N (1 /z , z , . . . , z N ) (3 . c )7ere c and c N are scalar functions to be specified later, and s = q k +2) is a parameterwhich determines the “level” k of the equation: here we consider the so-called level 1 case,namely with s = q .One can think of Eqs. (3 .
1) as an analogue of the quantum Knizhnik–Zamolodchikovequation ( q KZ) in the form introduced by Smirnov [17] (see also [18]), in which one re-places the periodic boundary conditions, implicit in the usual q KZ, with reflecting bound-aries [14]. More precisely, Eq. (3 . a ) is the exchange relation corresponding to the bulk,independent of boundary conditions, whereas Eqs. (3 . b, c ) implement the reflections atthe two boundaries.In [16], it was remarked that solving these equations for even size N = 2 n automat-ically provides a solution for odd size N − z N to zero(or equivalently to infinity). We therefore discuss in detail the case of even size now,postponing to Sect. 6 the discussion of the odd case. In [12], it was claimed that the system of equations (3 .
1) possesses a polynomialsolution of minimal degree 3 n ( n −
1) which is unique up to multiplication by a scalar. Toactually solve the equations (3 . N − . a ) froma triangular system with respect to the containment order of Dyck paths introduced inSect. 2.3. Indeed, when written in components, this equation reads: q − z i +1 − qz i z i +1 − z i ( τ i − π ( z , . . . , z n ) = X π ′ : π ′ = πe i π ′ = π Ψ π ′ ( z , . . . , z n ) (3 . τ i acts on functions of the z ’s by interchanging z i and z i +1 . Now consider the sumon the r.h.s.: it extends over the proper inverse images of π under e i . Picking π in theimage of e i (i.e. with an arch connecting points i and i + 1, as explained above), its inverseimages π ′ under e i all have dyck paths containing that of π (i.e. π ′ < π ) except one,say π ∗ , corresponding to the Dyck path of π with the box at position i removed, hencewith π < π ∗ . Hence Eq. (3.2) allows to express Ψ π ∗ in terms of only Ψ α , with α < π ∗ .The solution is therefore uniquely fixed by specifying the component corresponding to thesmallest link pattern π defined above, whose Dyck path that contains all others. The8atter is entirely fixed by the degree condition and factorization properties deduced fromthe q KZ system; the result is:Ψ π = Y ≤ i 4. Integral expressions for solutions of level q KZ The method introduced in [9] in order to obtain integral representations of Ψ was toexhibit a different basis than that of link patterns in which the integral expressions for thecomponents are relatively simple. We shall use the same basis here. Note that this sectionis “boundary conditions-independent” and its results are equally valid for say periodicboundary conditions.The elements of the basis we consider are indexed by strictly increasing integer se-quences of the form a ≡ { a , a , . . . , a n } , where 1 ≤ a i ≤ i − 1. We denote by O n theset of such sequences. The sequences in O n are in one-to-one correspondence with the linkpatterns in LP n in two different, inequivalent ways. One may indeed associate to each π ∈ LP n the sequence a i ( π ), i = 1 , , . . . , n , of the positions (counted from left to right,and taking values in { , , . . . , n − } ) of the openings of arches in π . Similarly, we mayassociate to π the sequence b i ( π ) = a i ( ρ ( π )), i = 1 , , . . . , n recording the closing positionsof the arches of π , counted from right to left, or equivalently the opening positions of thearches in the reflection ρ ( π ).In [9], we have constructed the change of basis from the link pattern basis to the“arch opening” basis, namely expressed the solutions Ψ π , π ∈ LP n in terms of multipleresidue integrals Ψ a ,...,a n , with { a , a , . . . , a n } ∈ O n . More precisely, Ref. [9] expressesany integral Ψ a ≡ Ψ a ,...,a n for weakly increasing sequences of a ’s as well, in terms of thesolution Ψ π in the link pattern basis, via the linear transformation:Ψ a ( z , . . . , z n ) = X π ∈ LP n C a ,π ( τ )Ψ π ( z , . . . , z n ) (4 . C a ,π ( τ ) expressed as follows. Let U k ≡ ( q k +1 − q − k − ) / ( q − q − ). For k ≥ U k is the Chebyshev polynomial of the second kind associated to − τ ,defined recursively by U k +1 = − τ U k − U k − with U = 1 and U − = 0. A given linkpattern π ∈ LP n may alternatively be thought of as a permutation of { , , . . . , n } withonly cycles of length 2. The arches forming π may therefore be described by the pairs( i, π ( i )), 1 ≤ i ≤ n , such that i < π ( i ), in which case i are the positions of the openingsand π ( i ) of the closings of the arches in π . Then we have the following formula: C a ,π ( τ ) = n − Y i =1 i<π ( i ) U µ ( a ,i ) (4 . µ ( a , i ) = card { j | i ≤ a j < π ( i ) } + i − π ( i ) − 12 (4 . π (with π ( i ) = i + 1), and replacing them with a factor U m − , where m is the total number of a ’s lying under that arch (namely such that a j = i ).The new link pattern thus obtained has one less arch, and its a ’s are relabeled accordingly,while m − a ’s are placed in position i − 1. The algorithm is iterated until the linkpattern becomes empty.If we moreover restrict the set of a ’s to O n , we get a true change of basis fromlink patterns to arch openings, in which C ( τ ) is a square invertible c n × c n matrix withpolynomial coefficients. The matrix C ( τ ) indeed enjoys the following properties. Let usfirst use the bijection between LP n and O n to write C a ,π ( τ ) ≡ C α,π ( τ ), where α ∈ LP n isuniquely determined by its arch opening positions a , a , . . . a n , counted from left to right.Let us moreover order the link patterns, say by increasing lexicographic order on the setsof positions of their arch openings. Then we have the property(P) C ( τ ) is a lower triangular matrix, with entries 1 on the diagonal, and polynomials of τ with integer coefficients elsewhere, and the same holds for C − ( τ ).Property (P) is easily derived as follows. First, it is clear that the diagonal terms C a ( π ) ,π =1, as µ ( a ( π ) , i ) = 0 for all arch openings i of π : indeed, the a ’a being the arch openings of π , the set { a j | i ≤ a j < π ( i ) } has one a per arch enclosed by the arch i → π ( i ), hencefortha total of ( π ( i ) − i + 1) / 2. Consequently, all the indices of the Chebyshev polynomialscontributing to (4.2) vanish, and as U = 1, the result follows. The triangularity is bestunderstood by following the abovementioned algorithm for constructing C . Indeed, at any11tep in the algorithm, the only cause for the matrix element to vanish is that one hasno a ’s under the little arch considered, as in this case one would get a factor U − = 0.The matrix element C α,π can only be non-zero if the arch openings of α occupy positionslexicographically larger that those of π . Indeed, by contradiction, assume the structure ofarch openings in α and π are the same up to say a position i where an arch opens in α while an arch closes in π . This means that the total number of arch openings for positions j > i is strictly larger in π than in α . Consequently, applying the above algorithm to thearches of π opening at positions > i , we see that at least one little arch in the process willhave no arch opening of α below it, thus receiving a weight U − = 0, and therefore thecorresponding matrix element C α,π vanishes.Finally, one can rewrite the q KZ equation itself using the linear combinations definedby Eq. (4.1). Note here that we are forced to use not only the components corresponding toour basis O N of increasing sequences, but also those corresponding to any non-decreasingsequence. In principle all of them can be reexpressed as linear combinations of increasingsequences only, but it is preferrable to avoid having to write these linear dependencerelations explicitly.All that is needed is the action of the e i on the Ψ a . We have the following Theorem 1: For any non-decreasing sequence a , . . . , a n such that the number i occursexactly k times, k ≥ , we have the formula: ( e i Ψ) a ,...,i,...,i |{z} k ,...,a n = U k − U k − Ψ a ,...,a n − U k − U k − (cid:0) Ψ a ,...,i − ,i,...,i |{z} k − ,...,a n + Ψ a ,...,i,...,i |{z} k − ,i +1 ,...,a n (cid:1) + U k − U k − Ψ a ,...,i − ,i,...,i |{z} k − ,i +1 ,...,a n (4 . (where for k = 0 the r.h.s. is zero). Proof: expand the l.h.s. in the basis of link patterns by using Eq. (4.1). We find:( e i Ψ) a = X π : π ( i ) = i +1 C a , e i ( π ) Ψ π + τ X π : π ( i )= i +1 C a ,π Ψ π (4 . e i its diagonal entries, equal to τ , andits non-diagonal entries, equal to 1.The Ψ π must be regarded here as independent objects, so that we must now checkEq. (4.4) for each link pattern π . This will be performed by a case by case analysis of thesituation around the sites ( i, i + 1). Each time only the coefficients involving the arches12tarting or ending at i, i + 1 differ from term to term in the equation, so that we can ignorethe remaining factors. The proof will be explained pictorially using the same conventionsas in appendix 1 of [9], that is by drawing the coefficient C a ,π as the usual (local) depictionof the link pattern π decorated by placing between sites i and i + 1 (inside a circle) thetotal number k of a ’s such that a j = i .There are 4 cases:(i) If i is an opening and i + 1 a closing of π , then π has a little arch ( i, i + 1): π ( i ) = i + 1.In this case the equality reduces pictorially to τ k i i+1 = U k − U k − k i i+1 − U k − U k − k-11 i i+1 − U k − U k − k-1 1 i i+1 + U k − U k − k-21 1 i i+1 or explicitly τ U k − = U k − U k − × U k − − U k − U k − × U k − + U k − U k − × U k − (4 . U k − U k − − τ = U k − U k − .In all other cases there is no little arch ( i, i + 1).(ii) If both i and i + 1 are openings, call p the total “weight” under the arch leaving i + 1,that is p = card { ℓ | i + 1 ≤ a ℓ < π ( i + 1) } − i +1 − π ( i +1) − , and q the remaining weightunder the bigger arch starting from i , excluding what is under the smaller arch andthe weight k under the segment [ i, i + 1), in order to make the pictorial descriptionsimpler: q = card { ℓ | π ( i + 1) ≤ a ℓ < π ( i ) } − π ( i ) − π ( i +1) − . Then the identity to proveis: k p q i i+1 = U k − U k − k p q i i+1 − U k − U k − k-11 p q i i+1 − U k − U k − k-1 p+1 q i i+1 + U k − U k − k-21 p+1 q i i+1 U k − U q − = U k − U k − × U p − U k + p + q − − U k − U k − × U p − U k + p + q − − U k − U k − × U p U k + p + q − + U k − U k − × U p U k + p + q − (4 . i and i + 1 are both closings is treated analogously.(iii) Finally, if i is a closing and i + 1 an opening, call p the weight under the arch ( π ( i ) , i )defined as before, and q the weight under the arch ( i + 1 , π ( i + 1)). Similary the proofof the identity p k q i i+1 = U k − U k − p k q i i+1 − U k − U k − p+1 k-1 q i i+1 − U k − U k − p k-1 q+1 i i+1 + U k − U k − p+1 k-2 q+1 i i+1 U k − U k + p + q − = U k − U k − × U p − U q − − U k − U k − × U p U q − − U k − U k − × U p − U q + U k − U k − × U p U q (4 . q KZ equation: general principle The idea to use integral representations for solutions of the q KZ equation is not newand there is a vast literature on the subject (cf the references in Sect. 11.2 of [19]). Weconsider here a very specific type of level 1 solutions, for which one expects a much simplerformula than generically. In the present context, this idea was used in [9] in the caseof the q KZ equation with the usual periodic boundary conditions. We now describe theprocedure in a slightly more general (boundary conditions-independent) setting.The idea is to define for any non-decreasing sequence ( a , . . . , a n ) the following quan-tity: Ψ a ,...,a n ( z , . . . , z N ) = Y ≤ i The function (4.9) solves the exchange relation of the q KZ equation, namelyit satisfies: ( t i Ψ) a ,...,a m − ,i,...,i |{z} k ,a m + k ,...,a n = U k − U k − Ψ a ,...,a m − ,i,...,i |{z} k ,a m + k ,...,a n − U k − U k − (cid:0) Ψ a ,...,a m − ,i − ,i,...,i |{z} k − ,a m + k ,...,a n + Ψ a ,...,a m − ,i,...,i |{z} k − ,i +1 ,a m + k ,...,a n (cid:1) + U k − U k − Ψ a ,...,a m − ,i − ,i,...,i |{z} k − ,i +1 ,a m + k ,...,a n (4 . t i acts only on the piecesof Ψ that are non-symmetric in ( z i , z i +1 ). When acting with t i on (4.9), we may restrictour attention to the non-symmetric part of the integrand. Secondly, we note that for anyfunction S ( u , . . . , u k ) satisfying the following vanishing antisymmetrizer property that A ( S ) ≡ X σ ∈ S k ( − σ S ( u σ (1) , . . . , u σ ( k ) ) = 0 (4 . I k ≡ I du · · · du k S ( u , . . . , u k ) Y ≤ ℓ 1. Wedecompose any permutation σ ∈ S k according to the image of 1, say σ (1) = m . Uponrelabelling indices, the corresponding permutation σ ′ is in S k − , and we may apply therecursion hypothesis to the summation over such σ ′ . We have A (cid:16) ∆ q ( u ) (cid:17) = k X m =1 X σ ∈ S k σ (1)= m ( − σ ∆ q ( u σ )= ( − k ( k − / k X m =1 k Y ℓ =1 ℓ = m qu m − q − u ℓ u m − u ℓ U U · · · U k − ∆( u ) (4 . q ( u , . . . , u m − , u m +1 , . . . , u k ) andreabsorbed the sign change by ( − k − into the ratio ∆( u ) / Q ℓ = m ( u m − u ℓ ). To conclude,we still have to prove the following sublemma: ϕ k ( u , . . . , u k ) ≡ k X m =1 k Y ℓ =1 ℓ = m qu m − q − u ℓ u m − u ℓ = U k − (4 . u , . . . , u k . To prove the latter, let us first note that itis a rational fraction, symmetric in the u ’s. Viewed as a function of u , it has possiblepoles at u , u , . . . , u k and is bounded at infinity. By symmetry, it is sufficient to computethe residue at u → u , for which only the two first terms in the summation contribute,leading to:Res u → u qu − q − u u − u k Y ℓ =3 qu − q − u ℓ u − u ℓ − qu − q − u u − u k Y ℓ =3 qu − q − u ℓ u − u ℓ ! = 0 (4 . ϕ k is bounded and has no pole in u , hence is independent of u , butas it is symmetric, it is a constant, say C k . To compute it, we take the limit u → ∞ ,and find the recursion relation ϕ k ( u , . . . , u k ) = q k − + q − ϕ k − ( u , . . . , u k ), henceforth C k = q k − + q − C k − . Moreover, by direct inspection, we find C = 0, therefore thesequence C is entirely fixed, and coincides with that of the Chebyshev polynomials of thesecond kind, namely C k = U k − = ( q k − q − k ) / ( q − q − ). This completes the proof of(4.17).Applying (4.17) to (4.16), and decomposing when necessary the permutation σ ac-cording to the images of 1 and/or k , say σ (1) = j and σ ( k ) = m , we arrive at A (cid:16) T ( u , . . . , u k ) (cid:17) ( − k ( k − / U · · · U k − ∆( u ) = ( qz i +1 − q − z i ) k Y ℓ =1 f ℓ g ℓ − ( qz i − q − z i +1 ) ! − ( z i +1 − z i ) ( U k − U k − − U k − k X j =1 f j k Y ℓ =1 ℓ = j qu j − q − u ℓ u j − u ℓ − U k − k X m =1 g m k Y ℓ =1 ℓ = m qu ℓ − q − u m u ℓ − u m + X ≤ j = m ≤ k f j g m qu j − q − u m u j − u m k Y ℓ =1 ℓ = j,m qu j − q − u ℓ u j − u ℓ qu ℓ − q − u m u ℓ − u m ) (4 . B in the following). To this end, we view it as a17ational fraction of the variable z i +1 , with possible poles at z i +1 → u s , ℓ = 1 , , . . . , k andat infinity. We first compute the residue at z i +1 → u s . From the definition (4.15), we haveRes z i +1 → u s ( g s ) = g ′ s = − ( q − q − ) u s , and all other terms have a finite limit, henceforth:1 g ′ s Res z i +1 → u s ( B ) = ( qu s − q − z i ) k Y ℓ =1 f ℓ k Y ℓ =1 ℓ = s qu ℓ − q − u s u ℓ − u s − ( u s − z i ) k X j =1 j = s f j Y ℓ =1 ℓ = s,j qu j − q − u ℓ u j − u ℓ − U k − k Y ℓ =1 ℓ = s qu ℓ − q − u s u ℓ − u s = ( u s − z i ) k Y ℓ =1 ℓ = s qu ℓ − q − u s u ℓ − u s k Y ℓ =1 ℓ = s f ℓ − k X j =1 j = s f j k Y ℓ =1 ℓ = s,j qu j − q − u ℓ u j − u ℓ + U k − (4 . k − u , . . . , u s − , u s +1 , . . . , u k ,its vanishing is actually the consequence of the following lemma, valid for all p ≥ p Y ℓ =1 f ℓ − p X j =1 f j p Y ℓ =1 ℓ = j qu j − q − u ℓ u j − u ℓ + U p − = 0 (4 . z i via the definition (4.15), the l.h.s. of (4.23) (which wedenote by D ) is a rational fraction, with possible poles at z i → q u ℓ , ℓ = 1 , , . . . , p and atinfinity. Let us first compute the residue at z i → q u s , for which the only contributionscome from Res z i → q u s ( f s ) = ( q − q − ) u s = f ′ s :1 f ′ s Res z i → q u s ( D ) = p Y ℓ =1 ℓ = s u ℓ − q u s q ( u ℓ − u s ) − p Y ℓ =1 ℓ = s qu s − q − u ℓ u s − u ℓ = 0 (4 . D has no finite pole, and it is moreover bounded at infinity, with limitlim z i →∞ ( D ) = q p − q p X j =1 p Y ℓ =1 ℓ = p qu j − q − u ℓ u j − u ℓ + U p − (4 . q p − qU p − + U p − = 0. Weconclude that D = 0, and henceforth B has no finite pole in z i +1 . We must now examinepossible residues at z i +1 → ∞ . The leading behavior of B when z i +1 → ∞ is polynomial18f degree ≤ 1. Noting that all lim z i +1 →∞ g ℓ = q − , the coefficient of z i +1 of B in this limitreads: B | z i +1 = q − + q − k k Y ℓ =1 f ℓ − k X j =1 q − f j k Y ℓ =1 ℓ = j qu j − q − u ℓ u j − u ℓ k X m =1 m = j k Y ℓ =1 ℓ = j,m qu ℓ − q − u m u ℓ − u m − U k − k X j =1 (cid:16) f j k Y ℓ =1 ℓ = j qu j − q − u ℓ u j − u ℓ + q − k Y ℓ =1 ℓ = j qu ℓ − q − u j u ℓ − u j (cid:17) + U k − U k − (4 . k → k − k in the second, thissimplifies into B | z i +1 = q − + q − k k Y ℓ =1 f ℓ − ( q − U k − − U k − ) k X j =1 f j k Y ℓ =1 ℓ = j qu j − q − u ℓ u j − u ℓ − U k − ( U k − − q − U k − )(4 . q − U k − − U k − = q − k and U k − − q − U k − = − q − k , we finally applythe lemma (4.23) with p = k , with the result B | z i +1 = q − − q − k U k − + q − k U k − = q − k ( q k − + U k − − qU k − ) = 0 (4 . B has no finite pole in z i +1 and is bounded at z i +1 → ∞ , it istherefore independent of z i +1 . We now evaluate B at z i +1 = 0, in which case all g ℓ = q ,and B | z i +1 =0 = − z i q + q k − k Y ℓ =1 f ℓ − ( qU k − − U k − ) k X j =1 f j k Y ℓ =1 ℓ = j qu j − q − u ℓ u j − u ℓ − U k − ( U k − − qU k − ) (4 . qU k − − U k − = q k − and U k − − qU k − = − q k , and we apply the lemma (4.23) with p = k to get B | z i +1 =0 = − z i ( q − q k − U k − + q k U k − ) = − z i q k ( q − k + U k − − q − U k − ) = 0 (4 . B = 0 identically, which implies that A ( T ) = 0 = A ( S ),which in turn implies (4.10), as explained above. This completes the proof of Theorem 2.19 .3. Case of reflecting boundary conditions All that has been described above can for example apply to the case of periodicboundary conditions treated in [9], avoiding the lengthy discussion found in this paper; inthis case the function F is just 1. In the present case of reflecting boundaries, the solutionto the q KZ equation must incorporate the new boundary conditions, cf Eqs. (3 . b, c ), whichcorrespond to a non-trivial choice of F , namely:Ψ a ,...,a n ( z , . . . , z N ) = Y ≤ i 1) = q n ( n − / ( q − q − ) n ( n − ( q + q − ) n ( n − / .Performing then the change of variables w i = − qu i − q − u i in the resulting expresion, weget: J a ,...,a n ( τ ) = τ n (2 n − I · · · I n Y m =1 du m πiu a m m " Q ≤ ℓ ≤ m ≤ n ( τ + ( τ − u ℓ + u m ) + τ ( τ − u ℓ u m ) Q nm =1 ( τ + ( τ − u m ) n × Y ≤ ℓ 5. Even case: proofs of various conjectures τ → τ → 0, the integral (4.33) is easily evaluated upon changing to variables v m = u m /τ and explicitly retaining only the leading terms when τ → K b ,...,b n ( τ ) ∼ τ n − P b i I · · · I n Y m =1 dv m πiv b m m Y ≤ ℓ 1) = β ( π ), which is also the number of boxes in thebox decomposition of the Dyck path associated to π .This result may now be immediately translated into an estimate for the small τ be-havior of the q KZ solution in the link pattern basis. Indeed, the change of basis (4.1)allows to identify Ψ π ( τ ) ∼ K b ( π ) ( τ ) when τ → 0. This is readily seen by writingΨ π ( τ ) = X α C − ( τ ) π,α K b ( α ) ( τ ) (5 . C − ( τ ) has all entries polynomial in τ , and that it is lower triangularwith respect to containment order of the Dyck paths associated to the link patterns, we22educe that any π such that C − ( τ ) π,α is non-zero must be contained in α , hence havea strictly smaller number of boxes if it is distinct from α . As K b ( α ) ( τ ) behaves like τ n ( n − − β ( α ) , we deduce that any contribution to the sum (5.3) with α = π is subleading,as β ( α ) < β ( π ).This completes the proof of the small τ conjecture of Ref. [7], in the case of even size,namely that Ψ π ( τ ) ∼ τ β ( π ) N (cid:16) b ( π ) , . . . , b n ( π ) (cid:17) (5 . b i ( π ) denote the positions of the arch closures of π , counted from right to left. τ For large τ , we obtain the leading contribution to K b ,...,b n ( τ ) by changing variables to v m = τ u m in the integral formula (4.33), and retaining only the leading order in τ withineach factor in the integrand. This yields K b ,...,b n ( τ ) ∼ τ P ( b i − I · · · I n Y ℓ =1 dv ℓ πiv b ℓ ℓ ∆( v ) n Y ℓ =1 (1 + v ℓ ) ℓ − (5 . K b ,...,b n ( τ ) ∼ τ P ( b i − det ≤ ℓ,m ≤ n (cid:18)I dv πiv v m − b ℓ (1 + v ) ℓ − (cid:19) ∼ τ P ( b i − det ≤ ℓ,m ≤ n (cid:18) ℓ − b ℓ − m (cid:19) (5 . π ( τ ), let us again consider the change of basis (5.3),and note that P ( b i ( π ) − 1) = n ( n − − β ( π ). We wish to prove that Ψ π ( τ ) ∼ K b ( π ) ( τ )at large τ . 210 1 2 i . . . h−1h ... Fig. 5: The strip decomposition of a typical Dyck path. To each ascendingstep ( i − , h − → ( i, h ) we associate a diagonal row of h − C α,π ( τ ) in τ is given by the following quantity. Definefirst h ( π, α ) as the sum over the arch openings of α of the total number of arches of π sitting above their position (an arch ( i, π ( i )) is said to sit above position j iff i ≤ j < π ( i )).The quantity h ( π, α ) is also the sum over the heights h i ( π ) in the Dyck path of π (orequivalently the position occupied by the path on the integer half-line at time i ), measuredat the positions i of the points in the Dyck path of α reached by an ascending step (i.e.such that h i ( α ) = h i − ( α ) + 1), namely: h ( π, α ) = n X i =1 h i ( α )= h i − ( α )+1 h i ( π ) (5 . h ( π, π ) = β ( π ) + n for any π ∈ LP n . Indeed, h ( π, π ) is the sum of heights of ends of ascending steps in the Dyck path of π . As illustratedin Fig.5, we may associate to each such ascending step ( i − , h i − ( π )) → ( i, h i ( π )) with h i ( π ) = h i − ( π ) +1 the diagonal strip of h i ( π ) − i + ℓ − , h i ( π ) − ℓ ), l = 1 , , ..., h i ( π ) − 1, and this exhausts all boxes of π . Such a “strip decomposition” wasconsidered in Ref. [20]. We deduce that β ( π ) = P i : h i ( π )= h i − ( π )+1 ( h i ( π ) − 1) = h ( π, π ) − n as there are exactly n ascending and n descending steps in the Dyck path. Then, usingthe definition (4.2) and the fact that the Chebyshev polynomials U m have degree m in τ ,we have d α,π ≡ deg (cid:16) C α,π ( τ ) (cid:17) = h ( π, α ) − h ( π, π ) (5 . The quantity f π ( α ) = h ( π, α ) + h ( α, α ) , where α runs over the link patterns whoseDyck path is included in that of π , reaches its maximum at α = π only. Proof: let us show that f π ( α ) is a non-decreasing function with the size of α , namely thenumber of boxes in the decomposition of its Dyck path. Assume α ′ ∈ LP n differs from α bya single box say at positions i − , i, i +1 in the Dyck path formulation, with identical heights h j ( α ′ ) = h j ( α ) + 2 δ j,i except at the position i . We see easily that h ( α ′ , α ′ ) = h ( α, α ) + 1as the ascending step ( i, i + 1) in α is replaced by ( i − , i ) in α ′ , and h i ( α ′ ) = h i +1 ( α ) + 1.Moreover, h ( π, α ′ ) = h ( π, α ) + h i ( π ) − h i +1 ( π ) ≥ h ( π, α ) − . h i ( π ) ≥ h i +1 ( π ) − 1. We deduce that f π ( α ′ ) ≥ f π ( α ), hence that f π ( π ) ≥ f π ( α ) for all α whose Dyck path is included in that of π . Finally, π is the unique maximum of f π ( α ),24s is easily seen by removing a box from π say at position i and comparing heights inthe Dyck paths. Indeed, we have say ( h i − ( π ) , h i ( π ) , h i +1 ( π )) = ( m, m + 1 , m ), and inthe box-removed π ′ we have ( h i − ( π ′ ) , h i ( π ′ ) , h i +1 ( π ′ )) = ( m, m − , m ). Hence h ( π, π ′ ) = h ( π, π ) − ( m + 1) + m and h ( π ′ , π ′ ) = h ( π, π ) − ( m + 1) + m , so that f π ( π ′ ) = f π ( π ) − d α,π (5.8): d α,π = h ( π, α ) − h ( π, π ) < h ( π, π ) − h ( α, α ) = β ( π ) − β ( α ). We may now prove that the large τ contribution to Ψ π ( τ ) is given by that to K b ( π ) ( τ ). This is done by induction on the(decreasing) number of boxes in α . Assume it is true for all π with a strictly largernumber of boxes than α . Then for all these π , Ψ π ( τ ) has degree equal to that of K b ( π ) ( τ ),namely n ( n − − β ( π ). Then the expression C α,π ( τ )Ψ π ( τ ), for α ⊂ π and π = α has degree d α,π + n ( n − − β ( π ) < n ( n − − β ( α ) by the above inequality. But K b ( α ) ( τ ) = P π C α,π ( τ )Ψ π ( τ ) has degree n ( n − − β ( α ), hence it must be attained by theterm π = α in the sum, and we deduce that Ψ α ( τ ) ∼ K b ( α ) ( τ ) for large τ . As the resultholds trivially for the largest link pattern π (the only non-vanishing matrix element of C with this first entry is just C π ,π ( τ ) = 1), this completes the desired proof. The fundamental remark of Ref. [9] for the even case was that summing Ψ a over aspecific subset of “opening arch” basis elements, namely the set of arch openings a ’s suchthat a i = 2 i − − ǫ i , ǫ i ∈ { , } , amounted to summing Ψ π over the whole set LP n . Thiswas readily seen as a property of the change of basis C α,π ( τ ) (4.2). Due to a reflectionsymmetry property, an analogous statement may be derived for the “arch closing” basiselements. As a result, the above sum rule for P π ∈ LP n Ψ π is obtained by summing theintegrals K b ,b ,...,b n ( τ ) over the b i = 2 i − − ǫ i , with ǫ i ∈ { , } .More precisely, let us consider K ( t | τ ) ≡ X ǫ i ∈{ , } t Σ ǫ i K − ǫ , − ǫ ,..., n − − ǫ n ( τ ) (5 . t = 0 corresponding to the “maxi-mum” component K , ,..., n − ( τ ) = Ψ π max ( τ ), where π max connects the points (2 i − , i )via little arches only (ii) t → ∞ corresponding to K , , ,..., n − ( τ ) which is also equal to asingle component, namely the link pattern π ′ max connecting (2 i, i + 1), and (1 , n ); (iii)25 = 1 corresponding to the sum rule P π ∈ LP n Ψ π = K (1 | τ ). From the definition (4.33), itis readily computed as K ( t | τ ) = I · · · I n Y m =1 du m (1 + tu m )2 πiu m − m Y ≤ ℓ ≤ m ≤ n (1 − u ℓ u m ) × Y ≤ ℓ 321 0 121 34 10(a) (b) (c)A CB Fig. 6: (a) The fundamental domain of a modified CSTCPP and its NILPdescription (blue and black paths) and (b) the correspondence (heights of thebox piles) to triangular arrays of integers (c). Colors are related to weights:red means a weight of τ , purple 1 /t , green x . With these conventions, (a)receives a weight τ × t − × τ (times a global factor of t n − = t ), while (b)and (c) have the weight x .In order to reconnect with the results of [7], it is convenient to rewrite K ( t | τ ), byshifting all indices: ℓ = j + 1, m = i + 1, r = s + 1, and noting that the first row/columndo not contribute to the determinant: K ( t | τ ) = det ≤ i,j ≤ n − "X s τ i +2 j − s (cid:18) i i − s (cid:19) (cid:18) tτ (cid:18) j j − s + 1 (cid:19) + (cid:18) j j − s (cid:19)(cid:19) (5 . t , the polynomials K ( t | τ ) correspond to a refined τ, t -enumeration ofCSTCPP △ , which can be described as follows. In the NILP formulation of the CSTCPP △ the CSTCPP △ are described by paths made by two orientations of lozenges in a fundamen-tal domain of the CSTCPP △ : these are the (colored) lozenges of types A, B on Fig. 6(a).In Ref. [7], these paths were viewed as pairs of paths sharing their arrival point, the pathson one side being one step longer than on the other side: the set of paths below the dashedline of Fig. 6(a) was identified as the NILP in bijection with TSSCPP (paths representedin black), while that above was viewed as an augmented one, with one more last step ineach path (paths represented in blue, with the last step taking place in the strip just abovethe dashed line).At each step of the paths, lozenges of type A above the diagonal dashed line and Bbelow (in red on the figure) are given a weight of τ in K ( t | τ ), except the last step of thelonger paths (in the strip just above the dashed line), where a factor t is given to lozengesof type B; it is however more convenient to consider for this last step that the factor of τ is replaced by 1 /t (purple lozenges on the figure) up to global multiplication by t n − .27et us now discuss various specializations of K ( t | τ ).As announced above, at t = 0, we find the maximal componentΨ π max ( τ ) = K (0 | τ ) = det ≤ i,j ≤ n − "X s τ i +2 j − s (cid:18) i i − s (cid:19)(cid:18) j j − s (cid:19) (5 . t → ∞ , we find:Ψ π ′ max ( τ ) = lim t →∞ t n K ( t | τ ) = det ≤ i,j ≤ n − "X s τ i +2 j − s +1 (cid:18) i i − s (cid:19)(cid:18) j j − s + 1 (cid:19) (5 . t = 1 the sum rule for the components of Ψ: X π ∈ LP n Ψ π ( τ ) = det ≤ i,j ≤ n − "X s τ i +2 j − s (cid:18) i i − s (cid:19) (cid:18) τ (cid:18) j j − s + 1 (cid:19) + (cid:18) j j − s (cid:19)(cid:19) (5 . t = τ − we find thegenerating function T n ( x = τ , 1) of [22]. The latter has several intepretations. One ofthem is the following: consider triangular arrays of non-negative integers a ij , i, j ≥ i + j ≤ n , with weakly decreasing rows and columns and such that a i ≤ n − i + 1 for all i .These arrays turn out to be in bijection with CSTCPP △ , see Fig. 6(b); to produce T n ( x, x to parts a ij such that a ij ≤ j − 1, see Fig. 6(c). Via the bijection thiscorresponds in terms of plane partitions to lozenges of type B that are below the diagonal.We now show that this is the same weight that is given to CSTCPP △ in K ( τ − | τ ). When t = τ − all red/purple lozenges get a weight of τ . Call n ab the number of lozenges of type a in region b where a = A, B, C and b = ↑ , ↓ corresponds to above/below the diagonal dashedline. Then the weight for CSTCPP △ is τ n A ↑ + n B ↓ , whereas the weight for triangular arraysis x n B ↓ . Now the number of tiles of each orientation is fixed (independent of the choice ofplane partition): in particular n A ↑ + n A ↓ = n ( n + 1) / − 1. Furthermore the number oftiles of the first 2 types in each region is also fixed: n A ↑ + n B ↑ = n ( n − / 2. We concludeimmediately that n A ↑ + n B ↓ = 2 n B ↓ + n − 1, which allows us to identify the weights takinginto account the prefactor t n − and the equality x = τ . T n ( x, 1) is also conjectured to be the x -enumeration of VSASM where a weight x isgiven to each pair of − 1s (conj. 3.2 of [22]). In the current framework there is no obviousexplanation of this coincidence. 28 . The odd case The link patterns in size 2 n + 1 are obtained bijectively from those in size one more2 n + 2, by simply erasing the rightmost arch and suppressing the rightmost endpoint2 n + 2, so that the opening point of the rightmost arch remains isolated and unmatched.The component of Ψ in size 2 n + 1 can then be obtained from the corresponding one insize 2 n + 2 by setting z N = 0. There is of course another bijection which consists on thecontrary in erasing the leftmost arch and endpoint, so that the closing point of the leftmostarch remains unmatched. This time setting z = ∞ allows to obtain Ψ in odd size 2 n + 1from even size 2 n + 2. Since the q KZ equation is not quite left-right symmetric these tworoutes lead to different expressions.Here we use the second route, having in mind that eventually we shall apply left-rightsymmetry as before to express everything in terms of closings instead of openings. Westart with Eq. (4.31) at N = 2 n + 2 with a = 1 and integrate over w ; we set z = ∞ atthis stage and obtain after shifting one step the indices of the a ’s, w ’s and z ’s:Ψ ′ a ,...,a n ( z , . . . , z N ) = Y ≤ i 1) of [22].Finally at t = τ we can identify K ′ ( τ | τ ) with the generating function T n ( τ , 0) definedin [22]. Once again, this is no surprise since T n ( x, 0) is the generating function for triangulararrays of non-negative integers a ij , i, j ≥ i + j ≤ n , with weakly decreasing rowsand columns and such that a i ≤ n − i for all i , which turn out to be in one-to-onecorrespondence with CSTCPP, see Fig. 7. As in the case of even size, one can check thatthe refinements are the same, i.e. that the weight x given to lozenges of one of the threetypes that are below the diagonal is the same weight that is given to CSTCPP in K ′ ( τ | τ )if one sets x = τ . Note that once again simple identities show that K ′ ( τ | τ ) is equal to K (0 | τ ), that is the component Ψ π max ( τ ) of size 2 n . 7. Conclusion In this paper, we have proved various conjectures regarding the minimal polynomialsolution of the q KZ equation with reflecting boundaries with q generic, expressed in the linkpattern basis. This was done by writing the solutions as multiple residue integrals, moduloa triangular change of basis. As both integrals and the change of basis are completelyexplicit, we therefore end up with an explicit formula for each component Ψ π ( z , ..., z N ).We hope these expressions will help us address the full Razumov–Stroganov conjecturewhich gives a conjectural interpretation for each Ψ π in the homogeneous case z = ... = z N = 1 (and q = − e iπ/ ), and hopefully come up with a more general combinatorialinterpretation of the polynomials Ψ π ( τ ) in the homogeneous case for generic q . Note that32e have now a numerical recipe for calculating the Ψ π ( τ ), via the explicit inversion of thechange of basis, and an explicit generation of the integrals.As stressed and proved in this paper, the above change of basis is independent ofthe details of the boundary conditions imposed in addition to the main exchange relation t i Ψ = ( e i − τ )Ψ. These details are simply reflected by the insertion of some specificsymmetric function F in the definition of the integrals. The techniques of the presentpaper may therefore presumably be adapted to include the other boundary conditionsconsidered in [23], parametrized by the root systems of classical Lie algebras.As shown in [8], generalizations of the Razumov–Stroganov sum rule have been ob-tained and proved in the case of the level 1 q KZ equation pertaining to higher rank ( sl k )algebras at specific values of the parameter q ( q = − e iπk +1 and q = − crossing loops consideredin [24,25,26], which is based on the Brauer algebra, and obtain integral formulae in thesame spirit as those of the present work. It would then be particularly interesting tounderstand their interrelation with the geometry of the Brauer loop scheme. Acknowledgments We acknowledge the support of European Marie Curie Research Training Networks“ENIGMA” MRT-CT-2004-5652, “ENRAGE” MRTN-CT-2004-005616, ESF program“MISGAM” and of ANR program “GIMP” ANR-05-BLAN-0029-01. We wish to thankJ. de Gier and P. 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