Towers of solutions of qKZ equations and their applications to loop models

Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear $q$-difference equations for $V_n$-valued meromorphic functions on a complex $n$-torus, with $V_n$ a module over the GL${}_n$-type extended affine Hecke algebra $\mathcal{H}_n$. The family $(\mathcal{H}_n)_{n\geq 0}$ of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms $\mathcal{H}_n\rightarrow\mathcal{H}_{n+1}$ the Hecke algebra descends of arc insertion at the affine braid group level. In this paper we consider qKZ towers $(f^{(n)})_{n\geq 0}$ of solutions, which consist of twisted-symmetric polynomial solutions $f^{(n)}$ ($n\geq 0$) of the qKZ equations that are compatible with the tower structure on $(\mathcal{H}_n)_{n\geq 0}$. The compatibility is encoded by so-called braid recursion relations: $f^{(n+1)}(z_1,\ldots,z_{n},0)$ is required to coincide up to a quasi-constant factor with the push-forward of $f^{(n)}(z_1,\ldots,z_{n})$ by an intertwiner $\mu_{n}: V_{n}\rightarrow V_{n+1}$ of $\mathcal{H}_{n}$-modules, where $V_{n+1}$ is considered as an $\mathcal{H}_{n}$-module through the tower structure on $(\mathcal{H}_n)_{n\geq 0}$. We associate to the dense loop model on the half-infinite cylinder with nonzero loop weights a qKZ tower $(f^{(n)})_{n\geq 0}$ of solutions. The solutions $f^{(n)}$ are constructed from specialised dual non-symmetric Macdonald polynomials with specialised parameters using the Cherednik-Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, $f^{(n)}$ coincides with the (suitably normalized) ground state of the inhomogeneous dense $O(1)$ loop model on the half-infinite cylinder with circumference $n$.


Introduction
Quantum Knizhnik-Zamolodchikov (qKZ) equations are consistent systems of linear q-difference equations that naturally arise in the context of representation theory of quantum affine algebras [14] and affine Hecke algebras [5]. They appear as consistency equations for form factors and correlation functions of various integrable models (see e.g., [19,26] for the first examples). In this paper we focus on Cherednik's qKZ equations associated to the GL n -type extended affine Hecke algebra H n . This case relates to integrable one-dimensional lattice models with quasi-periodic boundary conditions, with the integrability governed by the extended affine Hecke algebra H n . Important examples, also in the context of the present paper, are the XXZ spin-1 2 chain and the dense loop model.
The collection (H n ) n≥0 of extended affine Hecke algebras forms a tower of algebras with respect to algebra morphisms H n → H n+1 that arise as descendants of arc insertion morphisms B n → B n+1 for the groups B n of affine n-braids, cf. [1,3,15]. In this paper, we study families (f (n) ) n≥0 of solutions f (n) of qKZ equations taking values in H n -modules V n that are naturally compatible to the tower structure.
It leads us to introducing the notion of a tower (f (n) ) n≥0 of solutions of qKZ equations. The constituents f (n) of the tower are polynomials in n complex variables z 1 , . . . , z n , taking values in a finite dimensional H n -module V n . They are twisted-symmetric solutions of Cherednik's qKZ equations interrelated by the so-called braid recursion relations, meaning that f (n+1) (z 1 , . . . , z n , 0) coincides with the push-forward of f (n) (z 1 , . . . , z n ) by an H n -intertwiner µ n : V n → V n+1 up to a quasi-constant factor, where V n+1 is regarded as an H n -module through the tower structure of (H n ) n≥0 . In the terminology of [3], the collection {(V n , µ n )} n≥0 of H nmodules V n and H n -intertwiners µ n : V n → V n+1 is a tower of extended affine Hecke algebra modules. From this perspective, towers of solutions of qKZ equations are naturally associated with towers of extended affine Hecke algebra modules. The braid recursion relations are then determined by the module tower up to the quasi-constant factors.
In [3], the first and third authors constructed a family of module towers, called link pattern towers, which depends on a twist parameter v. The link pattern tower actually descends to a tower of extended affine Temperley-Lieb algebra modules. The representations V n are realized on spaces of link patterns on the punctured disc, which alternatively can be interpreted as the quantum state spaces for the dense O(τ ) loop models on the half-infinite cylinder (with n the circumference of the cylinder). The intertwiners µ n in the link pattern tower are constructed skein theoretically (for even n this goes back to [11]), and are in fact closely related to arc insertion morphisms in a relative version of the Roger-Yang [25] skein module in the presence of a pole (see [3,Rem. 8.11]). In this paper, we construct towers (f (n) ) n≥0 of solutions of qKZ equations relative to the link pattern tower with twist parameter one, and describe the corresponding quasi-constant factors in the braid recursion relations explicitly. We consider two cases.
We show that the (suitably normalized) ground states f (n) of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n form a tower of solutions relative to the link pattern tower. In this case, the associated affine Hecke algebra parameter is a third root of unity. This generalizes results from [11], where the braid recursion relations relating f (2k+1) (z 1 , . . . , z 2k , 0) to f (2k) (z 1 , . . . , z 2k ) were derived under the implicit additional assumption that a unique normalized ground state for the inhomogeneous dense O(1) loop model exists when one of the rapidities is set equal to zero (the latter is not guaranteed, since the transfer operator is no longer stochastic when one of the rapidities is set equal to zero). In an upcoming paper [2] the full set of braid recursion relations for the ground states is used to derive explicit formulas for various observables of the dense O(1) loop model on the infinite cylinder.
We generalize this example by constructing a tower of solutions (f (n) ) n≥0 for twist parameter one and for all values of the affine Hecke algebra parameter for which the loop weights of the associated dense loop model are nonzero. In this case, the constituents f (n) are constructed using the Cherednik-Matsuo correspondence [22,27]. The Cherednik-Matsuo correspondence, relating solutions of qKZ equations to common eigenfunctions of Cherednik's commuting Y -operators, can be applied in the present context since the link pattern modules are principal series modules, as we shall show in Theorem 6.6. It leads to the construction of the constituents f (n) of the tower in terms of nonsymmetric Macdonald polynomials. Subtle issues arise here since the two parameters of the associated double affine Hecke algebra satisfy an algebraic relation that breaks down the semisimplicity of the Y -operators. We resort to Kasatani's [20] work to deal with these issues. See [21] for an alternative approach to construct polynomial twisted-symmetric solutions f (n) of the qKZ equations using Kazhdan-Lusztig bases.
In both towers the constituent f (n) is a nonzero twisted-symmetric homogeneous polynomial solution of the qKZ equations of total degree 1 2 n(n − 1). In fact, this property characterizes f (n) up to a nonzero scalar multiple, a result that plays a crucial role in establishing the explicit braid recursion relations. In particular it allows us to prove the braid recursion relations for the suitably normalized ground states of the inhomogeneous dense O(1) loop models without addressing the issue of the existence of a unique normalized ground state when the rapidities are outside the stochastic regime.
The content of the paper is as follows. In Section 2 we recall the definitions of extended affine Hecke algebras and qKZ equations, and introduce the notion of a qKZ tower of solutions. In Section 3 we recall from [3] the definition of the link pattern tower. In Section 4 we determine necessary conditions for the existence of nonzero twisted-symmetric homogeneous polynomial solutions f (n) of total degree 1 2 n(n − 1) of the qKZ equations with values in the link pattern modules. We show that the existence implies that (f (n) ) n≥0 forms a tower of solutions relative to the link pattern tower, and we explicitly write down the corresponding braid recursion relations. The construction of the tower of solutions when the Hecke algebra parameter is a third root of unity is discussed in Section 5. The general case is discussed in Section 6. We derive a dual version of the braid recursion relations in Section 7. Lastly, in Appendix A we discuss uniqueness properties for various classes of twisted-symmetric solutions to qKZ equations, some of which were considered before in [8,11,21].
1.1. Acknowledgments. We thank Eric Opdam for a comment leading to the precise conditions on the affine Hecke algebra parameter for which the main theorem of the paper (Theorem 4.7) holds true. The work by Kayed Al Qasimi is supported by the Ministry of Education of the United Arab Emirates under scholarship number 201366644. Diagrams were coded using PSTricks.

Towers of solutions of qKZ equations
In this section we begin by recalling the extended affine Hecke algebra, the qKZ equations and introduce what we call a qKZ tower of solutions. The extended affine Hecke algebra can be defined using two different presentations. We make use of both presentations as one is more convenient for defining qKZ equations, while the other is more suitable for relating the algebra to the extended affine Temperley-Lieb algebra.
2.1. Extended affine Hecke algebras. Let t 1 4 ∈ C * . Definition 2.1. Let n ≥ 3. The extended affine Hecke algebra H n = H n (t 1 2 ) of type A n−1 is the complex associative algebra with generators T i (i ∈ Z/nZ) and ρ, ρ −1 and defining relations where the indices are taken modulo n. For n = 2 the extended affine Hecke algebra H 2 = H 2 (t 1 2 ) is the algebra generated by T 0 , T 1 , ρ ±1 with defining relations (2.1) but with the third relation omitted. For n = 1 we set H 1 := C[ρ, ρ −1 ] to be the algebra of Laurent polynomials in one variable ρ, and for n = 0 we set H 0 := C[X], the polynomial algebra in the variable X.
For n ≥ 1 the element ρ n ∈ H n is central.
For n ≥ 2 the affine Hecke algebra H a n = H a n (t 1 2 ) of type A n−1 is the subalgebra of H n generated by T i (i ∈ Z/nZ). For n ≥ 3 the first three relations of (2.1) are the defining relations of H a n in terms of these generators (for n = 2 the first two relations are the defining relations). Furthermore, H n is isomorphic to the crossed product algebra Z ⋉ H a n , where m ∈ Z acts on H a n by the algebra automorphism T i → T i+m (with the indices modulo n). Equivalently, m ∈ Z acts by restricting the inner automorphism h → ρ m hρ −m of H n to H a n . For n ≥ 2 the (finite) Hecke algebra of type A n−1 is the subalgebra H 0 n of H a n generated by T 1 , . . . , T n−1 . The defining relations of H 0 n in terms of the generators T 1 , . . . , T n−1 are given again by the first three relations of (2.1), restricted to those indices that they make sense.
Bernstein and Zelevinsky [23] obtained the following alternative presentation of the extended affine Hecke algebra (see also [18] for a detailed discussion).
Theorem 2.2. Let n ≥ 2 and define Y j ∈ H n for j = 1, . . . , n by The defining relations of H n in terms of these generators are given by , Note that ρ ∈ H n can be expressed as with respect to the Bernstein-Zelevinsky presentation of H n . Let A n be the commutative subalgebra of H n generated by Y ±1 1 , . . . , Y ±1 n . More can be said about the structure of H n in terms of the Bernstein-Zelevinsky presentation (see [23] and [18]) 1 , . . . , z ±1 n ] be a Laurent polynomial in n variables z 1 , . . . , z n . Let f = α∈Z n c α z α (c α ∈ C) be its expansion in monomials z α := z α1 1 · · · z αn n . Then, −→ A n of commutative algebras. In addition, the multiplication map is a linear isomorphism. In [3, §8] it was shown that there exists a unique unit preserving algebra map ν n : H n → H n+1 satisfying for n ≥ 2, ν n (T i ) = T i , i = 1, . . . , n − 1, satisfying ν 1 (ρ) = t − 1 4 ρT −1 1 for n = 1, and satisfying ν 0 (X) = t 1 4 ρ+t − 1 4 ρ −1 for n = 0. The ν n was obtained in [3, §8] as the Hecke algebra descent of an algebra homomorphism C[B n ] → C[B n+1 ], with B n the extended affine braid group on n strands, defined topologically by inserting an extra braid going underneath all the other braids it meets. At the end of this section, we require the algebra maps ν n in constructing towers of H n -modules and qKZ towers of solutions.

qKZ equations.
We consider Cherednik's [5,6] qKZ equations of type GL n . We will follow closely [27], and we will restrict attention to twisted-symmetric solutions of qKZ equations. The notations (m, k, ξ) in [27, §4.3] correspond to our (n, −t 1 2 , ρ). The qKZ equations depend on an additional parameter q, which we for the moment take to be an arbitrary nonzero complex number.
Recall that for n ≥ 1 and t 1 2 = 1, the extended affine Hecke algebra H n (1) is isomorphic to the group algebra C[W n ] of the the extended affine symmetric group W n ≃ S n ⋉ Z n . Writing s i (i ∈ Z/nZ) and ρ for the (Coxeter type) generators of W n , acting on C[z ±1 ] and C(z) := C(z 1 , . . . , z n ) by cf. Definition 2.1. Note that the W n -action on C[z ±1 ] is by graded algebra automorphisms, with the grading defined by the total degree. In addition, W n preserves the polynomial algebra C[z] := C[z 1 , . . . , z n ]. Define for n ≥ 1 and i ∈ Z/nZ, which we view as rational H n (t The key point in the construction of qKZ equations is the fact that for any H n (t 1 2 )-module V n with representation map σ n : H n (t 1 2 ) → End(V n ) and for q ∈ C * , the formulas define a left W n -action on the space V n (z) := C(z) ⊗ V n of V n -valued rational functions in z 1 , . . . , z n , where the W n -action in the right-hand side is the action on the variables as given by (2.4). For n = 0, we simply take ∇ = σ 0 acting on V 0 . The fact that (2.5) defines a W n -action is a consequence of the following identities for the R-operators R i (x), with the indices taken modulo n. The first equation is the Yang-Baxter equation [13,Vol. 5] in braid form. Note that in (2.4) and (2.5) the action of s 0 is determined by the action of s i (1 ≤ i < n) and of ρ, and hence does not have to be specified. We will often omit the explicit formula for the action of s 0 in the remainder of the paper. Following [27] we call the subspace V n (z) ∇(Wn) of ∇(W n )-invariant elements in V n (z) the space of twisted-symmetric solutions of the qKZ equations on V n . We need a more refined class of qKZ solutions, defined as follows.
If n ≥ 1 and Sol n (V n ; q, c) = {0}, then necessarily c ∈ C * . In this case denoting the vector space V n endowed with the twisted action σ c n : H n → End(V n ) defined by σ c n (T i ) := σ n (T i ) for i ∈ Z/nZ and σ c n (ρ) := c −1 σ n (ρ). We call c a twist parameter.
Here the basic representation π t − 1 2 ,q n acts on the first tensor where one needs to be well aware that the action on the variables through the basic representation is with respect to the extended affine Hecke algebra H n (t − 1 2 ) and the action on V n through σ n is with respect to the extended affine Hecke algebra H n (t 1 2 ). Before we can conclude this section with the introduction of the notion of a qKZ tower of solutions we need to establish some notation. Let A be a complex associative algebra and write C A for the category of left A-modules. Write Hom A (M, N ) for the space of morphisms M → N in C A , which we will call intertwiners. Suppose that η : A → B is a (unit preserving) morphism of C-algebras, then we write Ind η : C A → C B and Res η : C B → C A for the corresponding induction and restriction functor. Concretely, if M is a left A-module then with B viewed as a right A-module by b · a := bη(a) for b ∈ B and a ∈ A. If N is a left B-module then Res η (N ) is the complex vector space N , viewed as an A-module by a · n := η(a)n for a ∈ A and n ∈ N .
For a left H n+1 -module V n+1 we use the shorthand notation V νn n+1 for the left H n -module Res νn (V n+1 ). The following lemma introduces the concept of the module lift of a qKZ solution.
, which we still denote by µ n . Then, its restriction to Sol n (V n ; q, c n ) is a linear map µ n : Sol n (V n ; q, c n ) → Sol n (V νn n+1 ; q, c n ). Proof. This is immediate from the intertwining property Indeed, if f ∈ Sol n (V ; q, c n ) then it follows for n ≥ 1 from (2.9) that By the intertwiner µ n a qKZ solution f (n) (z) ∈ Sol n (V n ; q, c n ) gets lifted to a solution in Sol n (V νn n+1 ; q, c n ), taking values in the H n+1 -module V n+1 . Along with this upward module lift there is also a downward descent of a solution, which reduces the number of variables. It is defined as follows.
For n = 0 and f ∈ Sol 1 (V 1 ; q, c 1 ) we have By lifting solutions of qKZ equations by intertwiners µ n and descending solutions of qKZ equations by setting variables equal to zero we can connect qKZ solutions of different rank. This leads to the definition of a qKZ tower of solutions. The starting point is the following definition of a tower of extended affine Hecke algebra modules (compare with [3], where this notion was introduced for modules over extended affine Temperley-Lieb algebras, see also Section 3).
−→ · · · of extended affine Hecke algebra modules is a sequence {(V n , µ n )} n∈Z ≥0 with V n a left H n -module and µ n ∈ Hom Hn V n , V νn n+1 . To lift this notion of a tower to solutions of qKZ equations it is convenient to disregard quasiperiodic (with respect to the action of ρ) symmetric normalization factors h, i.e. polynomials h ∈ C[z] Sn satisfying ρh = λh for some λ ∈ C * . We call such h a λ-recursion factor, and λ the scale parameter. We write T n,λ ⊂ C[z] for the space of λ-recursion factors. Note that hf ∈ Sol n (V n ; q, λc n ) if f ∈ Sol n (V n ; q, c n ) and h ∈ T n,λ . By convention we define the space T 0,λ of λ-recursion factors for n = 0 to be C if λ = 1 and {0} otherwise.
If q is a root of unity, then we write e ∈ Z >0 for the smallest natural number such that q e = 1. We take e = ∞ if q is not a root of unity.
Proof. Let α ∈ Z n ≥0 . It suffices to show that β∈Snα z β ∈ C[z] Sn is a λ-recursion factor if and only if there exists a 0 ≤ m < e such that λ = q −m and α i ≡ m mod e for all i (where the latter condition for e = ∞ is read as α i = m for all i).
Note that ρ hence β∈Snα z β ∈ T n,λ if and only if λ = q −αi for all i = 1, . . . , n. This is equivalent to λ = q −m and α i ≡ m mod e for some 0 ≤ m < e.
The following lemma shows that by rescaling a nonzero symmetric polynomial solution of the qKZ equations by an appropriate recursion factor, it will remain nonzero if one of its variables is set to zero. Lemma 2.8. Let n ≥ 1 and let V n be a left H n -module with representation map σ n . If 0 = f ∈ Sol n (V n ; q, c n ) then there exists a unique m ∈ Z ≥0 and g ∈ Sol n (V n ; q, q m c n ) such that f (z) = (z 1 · · · z n ) m g(z) and g(z 1 , . . . , z n−1 , 0) ≡ 0.
Note that by Lemmas 2.4 and 2.5, we necessarily must have the compatibility condition between the twist and scale parameters in a qKZ tower of solutions (note that for n = 0 we have t

Extended affine Temperley-Lieb algebra
The qKZ towers we construct are built using modules of the extended affine Temperley-Lieb algebra, which is a quotient of H n . In this section we recall the definition of the extended affine Temperley-Lieb algebra and discuss the relevant tower of extended affine Temperley-Lieb algebra modules, following [3].
The extended affine Temperley-Lieb algebras arise as the endomorphism algebras of the skein category of the annulus, see [3] and references therein. We first give the definition of the extended affine Temperley-Lieb algebra in terms of generators and relations, and then discuss its relation to H n and the qKZ equations. For more details on the theory discussed in this section see [3] and references within. ) is the complex associative algebra with generators e i (i ∈ Z/nZ) and ρ, ρ −1 , and defining relations where the indices are taken modulo n. For n = 2 the extended affine Temperley-Lieb algebra ) is the algebra generated by e 0 , e 1 , ρ ±1 with the defining relations (3.1) but with the third relation omitted. For n = 1 we set T L 1 = H 1 = C[ρ, ρ −1 ], and for n = 0 we set The affine Temperley-Lieb algebra is the subalgebra T L a n of T L n generated by e i (i ∈ Z/nZ). The first three relations in (3.1) are the defining relations in terms of these generators (the first relation is the defining relation when n = 2). The (finite) Temperley-Lieb algebra is the subalgebra T L 0 n of T L a n generated by e 1 , . . . , e n−1 . The first three relations in (3.1) for the relevant indices are then the defining relations. Note that the dependence on the parameter t 1 2 of T L n is actually a dependence on t It is well known that for n ≥ 2 the assignments for i ∈ Z/nZ extend to a surjective algebra homomorphism ψ n : H n (t 1 2 ) ։ T L n (t 1 2 ) see e.g., [3,Prop. 7.2] and references therein. For n = 1 and n = 0 we take ψ n : H n → T L n to be the identity map.
Via the map ψ n the R-operators R i (x) := ψ n ( R i (x)) (i ∈ Z/nZ) on the extended affine Temperley-Lieb level are as rational T L n -valued function in x, with a(x) = a(x; t  2 ) given by Note that the R i (x) (i ∈ Z/nZ) satisfy the Yang-Baxter type equations (2.6) in T L n . The weights a(x) and b(x) will play an important role in the next section, where they appear as the Boltzmann weights of the dense loop model. We can now define the following analog of the qKZ solution space Sol n (V n ; q, c) (Definition By a slight abuse of notation we will denote this space of solutions again by Sol n (V n ; q, c). No confusion can arise, since Sol n (V n ; q, c) for the left T L n -module V n coincides with Sol n ( V n ; q, c), where V n is the H n -module obtained by endowing V n with the lifted H n -module structure with representation map σ n • ψ n .
From [3, Prop. 6.3] we have an algebra homomorphism I n : T L n (t for n ≥ 1. In particular, I n (ρ −1 ) = (t 1 4 e n + t − 1 4 )ρ −1 . Note that we have a commutative diagram is a tower of extended affine Temperley-Lieb modules if V n is a left T L n -module and µ n ∈ Hom T Ln V n , V In n+1 for all n ≥ 0. We sometimes write the tower as Note that (3.5) implies that an intertwiner µ n ∈ Hom T L n (V n , V In n+1 ) is also an intertwiner V n → V νn n+1 of the associated H n -modules. Hence, the tower {(V n , µ n )} n≥0 of extended affine Temperley-Lieb algebra modules gives rise to the tower {( V n , µ n )} n≥0 of extend affine Hecke algebra modules. Conversely, if {( V n , µ n )} n≥0 is a tower of extended affine Hecke algebra modules and the representation maps σ n : H n → End(V n ) factorize through ψ n , then the tower descends to a tower of extended affine Temperley-Lieb algebra modules. We will freely use these lifts and descents of towers in the sequel of the paper.
The tower of extended affine Temperley-Lieb modules relevant for the dense loop model is constructed from the skein category S = S(t 1 4 ) of the annulus, defined in [3]. We shortly recall here the basic features of the category S. For further details, we refer to [3, §3].
The category S is the complex linear category with objects Z ≥0 and with the space of morphisms Hom S (m, n) being the linear span of planar isotopy classes of (m, n)-tangle diagrams on the annulus A := {z ∈ C | 1 ≤ |z| ≤ 2}, with m and n marked ordered points on the inner and outer boundary respectively, modulo the Kauffman skein relation and the (null-homotopic) loop removal relation We consider here planar isotopies that fix the boundary of A pointwise. The ordered marked points on the boundary are (n) of algebras for n ≥ 0, with the algebra isomorphism θ n for n ≥ 1 determined by Moreover, in [3, Def. 6.1] an arc insertion functor I : S → S is defined using a natural monoidal structure on S. It maps n to n + 1 and, on morphisms, it inserts on the level of link diagrams a new arc connecting the inner and outer boundary while going underneath all arcs it meets (the particular winding of the new arc is subtle, see [3, §6] for the details). The resulting algebra homomorphisms I| EndS (n) : End S (n) → End S (n + 1) coincides with the algebra homomorphism I n by the identification of End S (n) with T L n (t Let v ∈ C * and set u := t The one-parameter family of link pattern towers of extended affine Temperley-Lieb algebra modules is now defined as follows (see [3, §10]). For The intertwiners φ n (n ≥ 0) are defined as follows. Consider the skein element The rather peculiar form of the intertwiners φ 2k−1 can be explained in terms of a Roger-Yang [25] type graded algebra structure on the total space Let D = {z ∈ C | |z| ≤ 2} and D * := D\{0}. A punctured link pattern of size 2k is a perfect matching of the 2k equally spaced marked points 2ξ i−1 2k (1 ≤ i ≤ 2k) on the boundary of D * by k non-intersecting arcs lying within D * . A punctured link pattern of size 2k − 1 is a perfect matching of the 2k marked points 2ξ j−1 2k−1 (1 ≤ j < 2k) and 0 by k non-intersecting arcs lying within D. Only the endpoints of the arcs are allowed to lie on {0} ∪ ∂D. Two link patterns are regarded the same if they are planar isotopic by a planar isotopy fixing 0 and the boundary ∂D of D pointwise. The arc connecting 0 to the outer boundary of D is called the defect line. An arc that connects two points on the boundary are sometimes referred to as an arch and an arch that connects two consecutive points that does not contain the puncture is called a little arch. We denote the set of punctured link patterns of size n by L n . As an example, the following punctured link patterns For twist parameter v = 1 we can naturally identify the nth representation space V n in the link pattern tower with C[L n ] as a vector space by shrinking the hole {z ∈ C | |z| ≤ 1} of the annulus to 0. The resulting action of T L n on C[L n ] can be explicitly described skein theoretically, see [3, §8].

qKZ equations on the space of link patterns
In this section we fix v = 1. We discuss the qKZ equations associated with the T L n -modules V n ≃ C[L n ] (n ≥ 0) from the link pattern tower, and we derive necessary conditions for the existence of qKZ towers of solutions. The existence of qKZ towers of solutions will be the subject of later sections. it was remarked that such solutions are uniquely determined by their fully nested component, and an explicit expression for the fully nested component was determined in case the solution is polynomial of total degree 1 2 n(n − 1) (existence of such a solution is a subtle issue). We recall these results here, extend them to qKZ solutions taking values in V In n+1 , and show how these results combined lead to explicit braid recursion relations.
where e i L ′ ∼ L means that L is obtained from e i L ′ by removing the loops in e i L ′ (there is in fact at most one loop). The coefficient γ if e i L ′ has a non null-homotopic loop, 1 otherwise.
Proof. This follows directly by rewriting the qKZ equations For the following lemmas concerning the uniqueness of solutions we need to impose that the loop weights −(t is a homogeneous polynomial of total degree m, then so is g Proof. In Appendix A we show by induction that, given g L∩ (z), the recursion relations (4.1) determine the other coefficients g (n) L (z) (L ∈ L n ) uniquely. For this the first equation in (4.1) is used in the following way: for L ′ ∈ L n and 1 ≤ i < n such that L ′ does not have a little arch between i and i + 1, denote by L ∈ L n the link pattern such that e i L ′ ∼ L, then g (n) L ′ (z) can be computed from other base components by the formula L ′ ,L = 0. By substituting the explicit expressions of the weights a(x) and b(x), this can be rewritten as L ′ (z) will be a homogeneous polynomial of total degree m if g L ′′ (z) are homogeneous polynomials of total degree m.
A similar result holds true for the restricted modules V In n+1 : ∈ L n+1 the fully nested diagram, then g (n) (z) = 0.
is a homogeneous polynomial of total degree m, then so is g Proof. The proof is similar to the proof of the previous lemma, but the check that the recursion relations coming from the qKZ equations for the representation V In,(c) n+1 determine all components in terms of the fully nested component g Then, g (n) L∩ (z) is a homogeneous polynomial of total degree m and (t 1 2 + 1)(t + 1) = 0, then m ≥ 1 2 n(n − 1) and C n (z) is a homogeneous symmetric polynomial of total degree m − 1 2 n(n − 1).
Proof. Note that L ∩ does not have a little arch connecting i and i + 1 for 1 ≤ i < n. By the recursion relation (4.1), it follows that The first result now follows immediately. For the second statement, suppose that g (n) L∩ (z) is a homogeneous polynomial of total degree m. Then, (4.2) and t 2 = 1 imply that g and the resulting quotient is invariant under interchanging z 1 and z 2 . One now proves by induction on r that g and the resulting quotient is symmetric in z 1 , . . . , z r . The second statement then follows by taking r = n.
It follows from the previous result that if the loop weights are nonzero and if there exists a nonzero g (n) ∈ Sol n (V n ; q, c n ) with coefficients being homogeneous of total degree 1 2 n(n − 1), then it is unique up to a nonzero scalar multiple and The following lemma is important in the analysis of qKZ towers of solutions relative to the link pattern tower {(V n , φ n )} n≥0 .
of φ n (L) in terms of the linear basis L n of V n . Then, c L,L (n+1) Proof. For n = 2k, consider a link pattern L ∈ L 2k that has a little arch connecting i, i + 1 for some i ∈ {1, . . . , 2k − 1}. All the link patterns in the image φ 2k (L) also contain the same little arch since the inserted defect line at the skein module level does not cross it (possibly after an appropriate number of applications of Reidemeister II moves). The only link pattern that does not contain a little arch connecting i, i + 1 for any 1 ≤ i < 2k is L ∩ . By the mapping φ 2k we have at the skein module level * and note that the image has k under-crossings. Resolving all the crossings using the Kauffman skein relations gives a linear combination of link patterns. The contribution to link pattern L ∩ ∈ L 2k+1 comes from taking the smoothing for each crossing . Each of these contributions gives a factor t − 1 4 , which establishes the result for n even. For the case n = 2k − 1 odd the first step of the argument is similar. The only link pattern that does not contain a little arch connecting i, i + 1 for any 1 ≤ i < 2k − 1 is L ∩ . By the mapping φ 2k−1 we have at the skein module level 2k * and note that each term in the image has k − 1 under-crossings. Resolving all the crossings using Kauffman's skein relations gives a linear combination of link patterns. The contributions to the link pattern L ∩ ∈ L 2k come from taking the smoothing for each crossing in the first term. Each of these contributions gives a factor t − 1 4 , which establishes the result for n odd.
The next lemma provides necessary conditions on the parameters q, c n for the existence of a qKZ tower of solutions of minimal degree relative to the link pattern tower.
, is a qKZ tower of solutions, with the associated braid recursion relations given by (4.3).
The proof of the theorem will be given in Section 6. The key step is the construction of g (n) (z) for generic t 1 4 ∈ C * in terms of specialised non-symmetric dual Macdonald polynomials using the Cherednik-Matsuo correspondence [27] and using results of Kasatani [20]. The generic conditions on t 1 4 can then be removed by noting that the constructed solution g (n) (z) is well defined over C(t 1 4 ) and the fact that the coefficients g The resulting qKZ tower of solutions (g (n) ) n≥0 from Theorem 4.7 is closely related to the inhomogeneous dense O(1) loop model on the half-infinite cylinder, see Section 5 and [11]. In fact, the constituents g (n) ∈ Sol n V n ; 1, 1 then are the renormalized ground states of the inhomogeneous O(1) dense loop models on the half-infinite cylinder. In this case, the braid recursion relations reduce to . . . , z 2k , 0) = (−1) k z 1 · · · z 2k φ 2k g 2k) (z 1 , . . . , z 2k )).

5.
Existence of solution for t 1 4 = exp(πi/3) In this section we recall the construction of the polynomial solutions g (n) (z) ∈ Sol n (V n (1); 1, 1) of degree 1 2 n(n − 1) for v = 1 and t 1 4 = exp(πi/3) (see Theorem 4.7). In this special case, the construction of the qKZ tower of solutions is facilitated by the fact that the underlying integrable model, the inhomogeneous dense O(1) loop model on the half-infinite cylinder, is stochastic. This allows one to construct g (n) (z) as a suitably renormalized version of the ground state of the inhomogeneous dense O(1) loop model, following [11].
The section begins with discussing the Temperley-Lieb transfer operator and then we specialize the analysis to the inhomogeneous dense O(1) loop model on the half-infinite cylinder. In this section v = 1. can be defined as follows [9,11]. For n > 0 consider the following two tiles which we denote by τ nw and τ ne , respectively, where 'nw' and 'ne' indicate that the north edge of the tile is connected to the west or east edge by an arc. Then, T (n) (x; z) = T (n) (x; z 1 , . . . , z n ) is defined by where τ i ∈ {τ nw , τ ne }, Note that the inner boundary of the annulus is always taken as the north edge of the tile. Moreover, for the case n = 1, tiling the annulus is done by stretching the tile so that the east and west edges are identified. The string of tiles covering the annulus can immediately be interpreted as an element in S n (t 1 4 ). Hence, by the algebra isomorphism θ n : T L n (t ). The case n = 0 is special. We define T (0) := θ 0 (X) (recall that T L 0 = C[X]). We also point out that since T L 1 = C[ρ, ρ −1 ] we have We will drop the isomorphism θ n when it is clear from context. Using diagrams we write the R-operator as and also as where we view the crossing in the annulus as a weighted sum of the two diagrams given in (5.1). Using the diagram description of the R-operator, the Yang-Baxter equations and inversion relation (lines 1 and 3 of (2.6)) can be depicted as The transfer operator can now be defined in terms of the R-operators, R i (x) for i ∈ Z/nZ, as follows. Let M (n) 0 (x; z) := ρR n−1 (x/z n )R n−2 (x/z n−1 ) · · · R 0 (x/z 1 ) ∈ T L n+1 be the monodromy operator where we view the auxiliary point as n + 1 ≡ 0 (modulo n + 1). Then, 0 (x; z) where cl 0 corresponds to the tangle closure [16] at the auxiliary point 0. In this specific case cl 0 amounts to disconnecting the two arcs from the inner-and outer boundary points labeled '0' and connecting them in End S (n) by an arc that under-crosses all arcs one meets.
The transfer operators with different values of x commute in T L n , This can be shown by interlacing two T operators with R-operators. In the literature, it is usually shown diagrammatically using the inversion relation and Yang-Baxter equation (5.2) of the R-operators. For an example of this technique, we refer the reader to [9] for dense loop models and [4] in general. Using the Yang-Baxter equation and the relations involving ρ (see (3.1)) one shows that In [11] the authors made the crucial observation that the R-operators R i (0), R i (∞) ∈ T L n can be interpreted as a single crossing in the skein description of the element, Consequently, Noting this over crossing and recalling the algebra map I n−1 : T L n−1 → T L n arising from the arc insertion functor, we obtain the following braid recursion relation for the transfer operator, which is due to [11, §2.4]:  [11,21]. We specialize in this section further to the case t . This means that all loops can be removed by a factor 1. As we shall discuss in a moment, the resulting O(1)-model is not only Bethe integrable but also stochastic. We identfy V n with C[L n ] as vector spaces (see the end of Section 3).
In [11] the authors stated the existence and uniqueness of a suitably normalized ground state of the inhomogeneous dense O(1) loop model, with z regarded as formal variables. For the convenience of the reader, we provide a full proof of this result. It uses the irreducibility and stochasticity of the transfer operator T (n) (x; z) for a particular parameter regime, and it uses the algebraic dependence of T (n) (x; z) on x and z.
Consider the matrix A (n) (x; z) := (A LL ′ (x; z)) L,L ′ ∈Ln of T (n) (x; z) with respect to the link pattern basis, The coefficients A LL ′ (x; z) depend rationally on x, z 1 , . . . , z n . For the special value t 1 4 = exp(πi/3) the Boltzmann weights a(x) and b(x) (see (3.3)) satisfy a(x) + b(x) = 1, hence L∈Ln A LL ′ (x; z) = 1 for all L ′ ∈ L n . Furthermore, we have 0 < a(x) < 1 if x = e iθ with 0 < θ < 2π/3; hence, A (n) (x; z) is left-stochastic if x/z j = e iθj with 0 < θ j < 2π/3 for j = 1, . . . , n. In this situation, A (n) (x; z) is irreducible; this follows from the fact that each L ∈ L n is a cyclic vector for the T L n -module V n , which can be proven as follows.
For n = 2k even, let L ln ∈ L 2k be the least-nested link pattern, which is the link pattern that has little arches connecting boundary points (2i − 1, 2i) for 1 ≤ i ≤ k such that the little-arches do not contain the puncture. All L ∈ L n can be mapped to L ln by acting with e 1 e 3 · · · e 2k−1 . In turn, L ln can be mapped to the fully nested link pattern L ∩ by the action of ρ k g k g k−1 · · · g 2 with g i := e i e i+2 · · · e 2k−i . Lastly, by the inductive argument in Appendix A, L ∩ can be mapped to any L ∈ L n . The case for n odd is analogous.
for all x ∈ C and such that L∈Ln g (n) Note that A (n) (z) is irreducible left-stochastic if z j = e −iθj with 0 < θ j < 2π/3; hence, for generic specialized values of the rapidities in this stochastic parameter regime, A (n) (z) has a onedimensional eigenspace with eigenvalue 1, spanned by the Frobenius-Perron eigenvector v F P (z), and the Frobenius-Perron eigenvector v F P (z) (normalized such that the sum of the coefficients is one), has the property that all its coefficients are positive. Hence, for generic values of the rapidities in the stochastic parameter regime, N (z) = 0. In particular, N (z) ∈ C(z)\{0}, and we may set g (n) (z) := L∈Ln g L (z) = 1. It follows from restricting to the stochastic parameter regime again that these two properties determine g (n) (z) uniquely.
Let x ∈ C and set Since L∈Ln A LL ′ (x; z) = 1 for all L ′ ∈ L n , we furthermore have L∈Ln g (n) L (x; z) = 1. Hence g (n) (x; z) = g (n) (z), i.e. T (n) (x; z) g (n) (z) = g (n) (z). This completes the proof of the uniqueness and existence of g n (z).
For the second statement, let 1 ≤ i < n and set h i (z) := R i (z i+1 /z i ) g (n) (s i z). Then, by the first formula of (5.3), , and the sum of the coefficients of h i (z) is one since a(x) + b(x) = 1. Hence h i (z) = g (n) (z), i.e., In the same way, one shows that ρ g (n) (z 2 , . . . , z n , z 1 ) = g (n) (z), now using the second equality of (5.3). This completes the proof of the lemma. Now we are ready to prove Theorem 4.7 in the special case that t Since in the present situation q = 1 and C n (z) is symmetric, we have that the renormalized function is also a symmetric solution of the qKZ equations, g (n) (z) ∈ C(z) ⊗ V n ∇(Wn) . Now g (n) (z) has fully nested component By Lemma 4.2 we conclude that g (n) (z) ∈ Sol n (V n ; 1, 1) is a homogeneous polynomial solution of total degree 1 2 n(n − 1), which completes the proof of Theorem 4.7 in the special case that t 1 4 = exp(πi/3).

Remark 5.3. From Proposition 5.1 it follows immediately that
I n ( T (n) (x; z 1 , . . . , z n ))g (n+1) (z, 0) = g (n+1) (z, 0). when t 1 4 = exp(πi/3). In [11] the authors use this equation to prove the braid recursion relation for v = 1, 1 = q t 1 4 = exp(πi/3) and n even (see Remark 4.8). However, they implicitly assume that g (n+1) (z, 0) is uniquely characterized as ground state of T (n+1) (1; z, 0), which is though not clear since there is no stochastic parameter regime when one of the rapidities is set equal to zero. We have circumvented this problem here by using the characterization of g (n) (z) as a twisted symmetric solution of qKZ equations. 6. Existence of solutions for generic t 1 4 In this section we construct for generic t A major difference between the generic case and the case that t 1 4 = exp(πi/3) is that we do not have the argument of a stochastic matrix to construct g (n) using the Frobenius-Perron theorem.
We instead use the Cherednik-Matsuo correspondence [27]. This is different from the approach in [21], where Kazhdan-Lusztig bases are used.
In order to be able to apply the Cherednik-Matsuo correspondence, we first need to identify the link pattern representations V n with principal series representations. This is done in the first subsection, for general twist parameter v. In the subsequent subsection, we recall the Cherednik-Matsuo correspondence and rephrase it in terms of dual Y -operators. In the last subsection, we prove Theorem 4.7 by constructing the polynomial solution of the qKZ equation from dual nonsymmetric Macdonald polynomials with specialised parameters.
For fixed v ∈ C * the link patterns L n form a (non-canonical) basis of V n . We can naturally identify V n with C[L n ] as a vector space by shrinking the hole {z ∈ C | |z| ≤ 1} of the annulus to 0. A choice needs to be made for the winding of the defect line, unless v = 1.
6.1. V n as a principal series module. In this section we take n ≥ 2, and we fix v ∈ C * . We recall first the definition of the principal series representation M I (γ) of the affine Hecke algebra H n = H n (t 1 2 ). Let ǫ i (1 ≤ i ≤ n) denote the standard basis of R n . Set R 0 := {ǫ i − ǫ j |1 ≤ i = j ≤ n}, the root system of type A n−1 . We take R + 0 := {ǫ i − ǫ j |1 ≤ i < j ≤ n} the set of positive roots. The corresponding simple roots are α i := ǫ i − ǫ i+1 (1 ≤ i < n). We write s α (α ∈ R 0 ) for the reflection in α. Then, the simple reflections s i := s αi (1 ≤ i < n) correspond to the simple neighboring transpositions ] and H n , respectively.
For I ⊆ {1, . . . , n − 1}, we write For γ ∈ T I let χ I γ := H I (t  Let S n,I = s i | i ∈ I ⊆ S n be the standard parabolic subgroup generated by the simple neighboring transpositions s i (i ∈ I), and S I n the minimal coset representatives of S n /S n,I . For w ∈ S n , let T w ∈ H 0 n be the element T w = T i1 T i2 · · · T ir if w = s i1 s i2 · · · s ir is a reduced expression. This is well defined since the T i 's satisfy the braid relations. A linear basis of M I (γ) is given by {v I w (γ) := T w ⊗ HI 1 χ I γ } w∈S I n . For a finite-dimensional left H n -module V and ξ ∈ (C * ) n , we define the subspace of vectors of weight ξ by For 1 ≤ i < n set (6.1) The following theorem is well known, see [27,Thrm. 2.8,Cor. 2.9] and references therein. Theorem 6.1. For w ∈ S n and w = s i1 s i2 · · · s ir a reduced expression, If V is a left H n -module, then the previous theorem implies that I w (V ξ ) ⊆ V wξ for w ∈ S n and ξ ∈ (C * ) n .
It is known that M I (γ) is calibrated for generic γ ∈ T I with corresponding weight decomposition with b I w (γ) := I w ⊗ HI 1 χ I γ (see e.g., [27,Prop. 2.12], for the specific additional conditions on γ). We now view the T L n -module V n from the link pattern tower as an H n -module through the surjective algebra map ψ n : H n ։ T L n satisfying ψ n (T i ) = e i + t − 1 2 (1 ≤ i < n) and ψ n (ρ) = ρ. The aim is to show that V n is isomorphic to M I (γ) for an appropriate subset I ⊆ {1, . . . , n − 1} and γ ∈ T I for generic t 1 4 . As a first step, we create explicit weight vectors in V n . Write k = ⌊ n 2 ⌋ and let J ⊆ {1, . . . , k} be a subset, say J = {j 1 , . . . , j r }, 1 ≤ j 1 < · · · < j r ≤ k. Let D n J be the element in V n shown in Figure 1. Note that in the definition of D n J , the arches (2m − 1, 2m) include the hole of the annulus if m ∈ J, and (2j s+1 − 1, 2j s+1 ) is positioned over (2j s − 1, 2j s ). Furthermore, D 2k+1 J is obtained from D 2k J by inserting the defect line at 2k + 1, which is positioned over all other paths.
We require the skein theoretic description of ψ n (Y j ) ∈ T L n . From the expression Y j = with Y j ∈ T L n ≃ End S (n) the skein class of In particular we have Q n (η) ∈ V n,ξ(η) with weight Proof. It suffices to show that Y j Q n (η) = ξ j (η)Q n (η). Write k := ⌊n/2⌋. There are three cases to consider, j = 2i, 2i − 1 (for 1 ≤ i ≤ k) and, if n is odd, j = n = 2k + 1. We consider first j = 2i. Note that by the definition of D n J an arch is placed on top of the previous arch if they both encircle the hole of the annulus. For Y 2i D n j , the path connected to 2i that is wound around the diagram passes over all paths connected to l < 2i and under all paths connected to l > 2i. Due to these properties, the action of Y 2i on D n J will only affect the arch (2i − 1, 2i) and leave the others unchanged.
Consider now Y 2i Q n (η) and combine the terms J and J ∪ {i} for subsets J not containing i, Focusing on the action of Y 2i on the terms in the bracket, we claim that To prove the claim we show in V 2k+1 (note that the defect line creates a subtle difference) and the third for Y 2i D n J∪{i} in V n (in this case, the defect line does not affect the calculation): is analogous. The proof that Y 2k+1 Q 2k+1 (η) = vQ 2k+1 (η) with n = 2k + 1 odd is simpler. All that the operator Y 2k+1 does is wind the defect line a full turn around the hole of the annulus. The operator keeps the defect line above all other curves. This full turn in V 2k+1 can then be removed by the multiplicative factor v.
From now on we choose η = (v ǫn , . . . , v ǫn ) and we write Q n , ξ and ξ for the corresponding Q n (η), ξ(η) and ξ(η). Concretely, if n odd. Lemma 6.3. Q n = 0 for generic t 1 4 . Proof. Consider first n = 2k even. Let Z 2k ∈ Hom S (2k, 0) be the skein class of the (2k, 0)-link diagram with little arches connecting 2i − 1 and 2i for 1 ≤ i ≤ k. Composing on the left with Z 2k defines a linear map Hom S (0, 2k) → End S (0) that descends to a well-defined linear map which is a nonzero Laurent polynomial in t 1 4 (look at its highest order term). For n = 2k + 1 odd, we apply a similar argument, now using the element Z 2k+1 ∈ Hom S (2k + 1, 1) which is the skein class of the (2k + 1, 1)-link diagram with little arches connecting the inner boundary points 2i − 1 and 2i (1 ≤ i ≤ k) and with a defect line connecting the inner boundary point at 2k + 1 to the outer boundary point at 1. Then, the resulting linear map for some κ ∈ Z, which again is a nonzero Laurent polynomial in t To establish an identification V n ≃ M I (γ) as H n -modules, we will use Q n and the intertwiners I w (w ∈ S n ) to construct the corresponding cyclic vector in V n . But first we determine what the subset I ⊆ {1, . . . , n − 1} should be.
Set I (n) := {1, . . . , ⌈n/2⌉ − 1, ⌈n/2⌉ + 1, . . . , n − 1}. The associated parabolic subgroup S n,I (n) of S n is isomorphic to S k × S k if n = 2k even, and S k × S k−1 if n = 2k − 1 is odd. Lemma 6.4. Dim(V n ) = #(S n /S n,I (n) ) = #S I (n) n . Proof. For n = 2k even, L 2k is in bijective correspondence with the set of binary words of length 2k with letters α, β of length 2k such that k letters are α. The bijection is as follows. Orient the outer boundary of the punctured disc anticlockwise. Given L ∈ L 2k , orient the arcs in L in such a way that the closed oriented loop obtained by adding a piece of the oriented outer boundary of the punctured disc, is enclosing the puncture. Then, the word of length 2k in the letters {α, β} is obtained by putting α as the ith letter if the orientation of the arc at i is away from i, and β if it is towards i.
In the odd case n = 2k − 1 we create a bijective correspondence of L 2k−1 with the set of binary words of length 2k − 1 with letters α, β such that k letters are α by a similar procedure, with the only addition that α is assigned to the outer boundary point that is connected to the puncture.
Clearly the cardinality of the set of such binary words is equal to #(S n /S n,I (n) ).
We define w n ∈ S n as follows, Note that w 2k = ι 2k−1 (w 2k−1 ) with ι n : S n−1 ֒→ S n the natural group embedding extending σ ∈ S n−1 to a permutation of {1, . . . , n} by σ(n) = n. Note that It follows that w −1 n ∈ S I (n) n , cf. Remark 6.5. We now define γ = γ (n) ∈ (C * ) n by γ := w n ξ ∈ (C * ) n with ξ the weight of V n as given by (6.2). Concretely we have Note that γ ∈ T I (n) , and hence we have the associated principal series module M I (n) (γ).
Theorem 6.6. For generic t where e is the unit element of the symmetric group S n .
Proof. We have I wn Q n ∈ V n,γ by Lemma 6.2 and Theorem 6.1. Furthermore, by Theorem 6.1 again, I w −1 n I wn Q n = e wn (ξ)Q n and e wn (ξ) = for generic t 1 4 , since R + 0 ∩ w −1 n R − 0 consists of the roots ǫ 2l − ǫ 2m−1 (l < m). Hence, I wn Q n = 0 for generic t 1 4 . Consider now the vectors Then, for w ∈ S I (n) n we have I w −1 u w = e w (γ)I wn Q n by Theorem 6.1. Furthermore, e w (γ) = 0 for generic t 1 4 since for w ∈ S I (n) n we have n , in view of Remark 6.5. It follows that 0 = u w ∈ V n,wγ for all w ∈ S I (n) n . Hence, by Lemma 6.4, for generic t with V n,wγ = Cu w for all w ∈ S I (n) n , since the wγ's (w ∈ S I (n) n ) are pairwise different for generic t 1 4 . It remains to show that T i u e = t − 1 2 u e for i ∈ I (n) and generic t 1 4 . Fix i ∈ I (n) . Then I i u e ∈ V n,siγ = {0} since s i γ ∈ {wγ | w ∈ S I (n) n } for generic t 1 4 . By the explicit expression of the intertwiner I i (see (6.1)), we then obtain Hence, T i u e = t − 1 2 u e , as desired.
6.2. The Cherednik-Matsuo correspondence. Now that we have identified the link pattern modules V n with principal series representations, we can apply the Cherednik-Matsuo correspondence to analyse the existence of polynomial solutions of the associated qKZ equations. The Cherednik-Matsuo correspondence gives a bijective correspondence between meromorphic twisted symmetric solution to qKZ equations associated with a principal series module and suitable classes of meromorphic common eigenfunctions for the action of the Y -operators under the basic representation [22,27].
The version of the Cherednik-Matsuo correspondence we need is as follows. If I ⊆ {1, . . . , n−1} and w ∈ S n , then we write w ∈ S n,I and w ∈ S I n for the unique elements such that w = ww. Let w 0 ∈ S n be the longest Weyl group element, mapping j to n + 1 − j for j = 1, . . . , n. Let Theorem 6.7. Fix c ∈ C * , I ⊆ {1, . . . , n − 1} and ζ ∈ T I . Then, we have a linear isomorphism Proof. For c = 1, this is an easy consequence of [27,Cor. 4.4 & Thm. 4.14]. For general c, it then follows using the fact that M I (c −1 ζ) ≃ M I (ζ) (c −1 ) with isomorphism given by v I w (c −1 ζ) → v I w (ζ) for w ∈ S I n , and Sol n (M I (ζ) (c −1 ) ; q, 1) = Sol n (M I (ζ); q, c).
For generic q and t 1 4 (or indeterminates), the monic dual nonsymmetric Macdonald polynomial ] of degree λ ∈ Z n is the unique Laurent polynomial satisfying the eigenvalue equations such that the coefficient of z λ in the expansion of E λ in monomials {z ν } ν∈Z n is one. It is well known that E λ is homogeneous of total degree |λ| := λ 1 + · · · + λ n . In addition, E λ ∈ C[z] if and only if λ ∈ Z n ≥0 . The intertwiners with respect to the dual Y -operators are defined by cf. [20, Lemma 2.6]. Then, for 1 ≤ i < n, Kasatani [20] analyzed the dual nonsymmetric Macdonald polynomials E λ with parameters specialized to t −k−1 q r−1 = 1 with 1 ≤ k ≤ n − 1 and r ≥ 2. In our situation, we are going to need the special case that k = 2 and r = 3, i.e., when t −3 q 2 = 1 (cf. Theorem 4.7). In fact, for our purposes it suffices to take q = t 3 2 . We recall some key results from [20] in this special case. Definition 6.10. We say λ ∈ Z n has a neighbourhood if it has a pair of indices (i, j) such that condition 1 and 2 are satisfied: By [20,Thm. 3.11], the dual non-symmetric Macdonald polynomial E λ can be specialized at q = t 3 2 if λ ∈ B (2,3) . For q = t such that z ia+1 = z ia tq ra for a = 1, 2, r 1 + r 2 ≤ 1, and i a < i a+1 if r a = 0}, and define the ideal I (2,3) ⊆ C[z] by Then, for q = t by [20,Thm. 3.11].
With the last proposition we have completed the proof of Theorem 4.7 for generic t 1 4 ∈ C * (note that Lemma 4.2 implies uniqueness, and that for n = 1 the desired unique solution g (1) is simply given by the constant function g (1) ≡ 1). The remark following Theorem 4.7 then completes the proof of Theorem 4.7 for all values t 1 4 ∈ C * for which (t 1 2 + 1)(t + 1) = 0.
As in Lemma 4.2, it follows that g (n) (z), as symmetric solution of these qKZ equations, is determined by the fully nested component and that all coefficients g (n) L (z) are homogeneous polynomials in z 1 , . . . , z n of total degree 1 2 n(n − 1). In particular, ρf (z 2 , . . . , z n , qz 1 ) = d n f (z).
Using slightly modified versions of Lemmas 2.4 and 2.5 one now shows that . The first two cases have been considered before in the literature, see [8,11,21]. We recall these here in detail, since the technicalities play an important role in proving the most delicate third case.
A link pattern of size 2k is a diagram with 2k equally spaced points on the boundary of the disc D that are connected by k non-intersecting curves lying within the disc. To establish convention the points are numbered 1 to 2k going counter-clockwise around the disc. We denote the set of link patterns of size 2k by LP 2k . As an example, LP 6 consists of the following link patterns: Link patterns can also be drawn by placing the endpoints on a horizontal line such that the k nonintersecting curves lie above it. To establish convention, the points are numbered in increasing order from left to right. As an example the link patterns of LP 6 can be drawn as respectively. Due to this form the curves are sometime referred to as arches and a little arch is one that connects two consecutive points. A Dyck path of length 2k is a lattice path from (0, 0) to (2k, 0) with steps (1, 1) called a rise and (1, −1) called a fall, which never falls below the x-axis. We denote the set of Dyck paths of length 2k by DP 2k . As an example DP 6 consists of the following Dyck paths: A Dyck path can also be encoded by a string of 2k numbers (a 1 , . . . , a 2k ) where a j for 1 ≤ j ≤ 2k is the height of the path after step j. Furthermore, for a Dyck path L we define its content |L| to be the number of boxes within the gray triangle that lie above the path. For example, for the last Dyck path L ∈ DP 6 in the example above, |L| = 3. There exists a bijection between LP 2k and DP 2k . To go from link patterns to Dyck paths, consider the link pattern drawn on a horizontal line and traverse along the line from left to right. Each point i that is the beginning/end of an arch corresponds to a rise/fall at step i in the Dyck path. To go from Dyck paths to link patterns, for each rise draw the start of an arch and for each fall an end, and then complete the diagram by connecting a start with an end such that the arches do not intersect. For example, the ordered lists of elements in LP 6 and DP 6 are the same under the bijective correspondence.
The bijection allows us to establish a containment ordering on link patterns. For two link patterns L, L ′ ∈ LP 2k , we say that L contains L ′ if the entire corresponding Dyck path of L ′ can be drawn along or below the Dyck path of L. More formally, let L and L ′ correspond to the Dyck paths (a 1 , . . . a 2k ) and (b 1 , . . . b 2k ), respectively. Then, L contains L ′ if a j ≥ b j for all 1 ≤ j ≤ 2k. As an example, in the list of Dyck paths in DP 6 the first path contains all other paths. Furthermore, note that if L contains L ′ then |L| ≤ |L ′ |.
Using the disc diagrams, one can define the action of T L 2k on C[LP 2k ] similarly to that on C[L 2k ], one just ignores the puncture so we do not have the loop removal rule for non-contractible loops. Using the horizontal line diagrams is more suitable when discussing the action of the finite Temperley-Lieb algebra T L f 2k . The induced action of T L f 2k on C[DP 2k ] can then be described as follows.
At step i for 1 ≤ i ≤ 2k − 1, a Dyck path can have one of three different local situations; (1) Steps i, i + 1 form a local maximum, i.e. a rise followed by a fall; (2) Steps i, i + 1 form a local minimum, i.e. a fall followed by a rise; (3) Steps i, i + 1 form a slope, i.e. two consecutive rises or falls.
If steps i, i + 1 form a local maximum, then the action of e i acts as a scalar, leaving the path unchanged and multiplying by a factor −(t 1 2 + t − 1 2 ) (line 1, Figure 2). If steps i, i + 1 form a local minimum, then e i changes it into a local maximum (line 2, Figure 2). For a slope, if it is two consecutive rises, say with heights a i = m and a i+1 = m + 1, then let j > i + 1 be the first step that is a fall with a j = m. The action of e i then changes step i + 1 into fall and j into a rise, creating a local maximum at i, i + 1 and decreasing the height of the path between i and j by two (line 3, Figure 2). This decrease in height shifts the internal path down, and we refer to it as a collapse. If the slope is downwards with height a i = m, a i+1 = m − 1, let j < i be the last rise with a j = m + 1. Then, the action of e i changes step i and j to a rise and fall, respectively. This creates a local maximum at i, i + 1 and causes a collapse decreasing a l by two for j ≤ l < i.
Note that a collapse leads to a smaller Dyck path in the inclusion order. Figure 2 gives a diagrammatic definition of the action on Dyck paths. The dotted frame indicates the section of the paths where they differ and the dotted line in the third mapping represents a Dyck path of length j − i − 2. The case for two consecutive falls is the same as the third line but with the diagrams reflected across a vertical line in the middle of the diagrams.
A.1. Link patterns. Let L 0 ∈ DP 2k denote the Dyck path with k rises followed by k falls i.e. (1, 2, . . . , k, k − 1, . . . , 0). Note that L 0 contains all Dyck paths in DP 2k and |L 0 | = 0. We show that the solution g (2k) (z) ∈ Sol 2k (C[LP 2k ]; q, c 2k ) is determined by its base component g L (z) ≡ 0 for all L ∈ DP 2k . This is done using the qKZ equations (4.1), which give for 1 ≤ j < 2k, where e j L ′ ∼ L means that L is obtained from e j L ′ by removing the loops in e j L ′ (there is in fact at most one loop). The coefficient γ has a null-homotopic loop,  1 otherwise.
We begin with the inductive hypothesis, . Now consider a Dyck path L such that |L| = m with a local maximum at, say, step i with a i > 1 (if such a local maximum does not exist, then L is the unique Dykh path with maximal content |L|, and hence there is nothing to prove). We use equation (A.1) for j = i and examine the pre-images L ′ in the sum on the right-hand side. We find that besides L there is only one pre-image that is contained by L. This is the pre-image that has a local minimum turned into a local maximum by the action of e i . Let us denote this particular Dyck path by N . Switching a local minimum to a local maximum is equivalent to removing a box, so we have |N | = m + 1. Furthermore, all other pre-images L ′ = N contain L so |L ′ | ≤ m and g (2k) N (z) ≡ 0 with |N | = m + 1 by (A.1). Since |L 0 | = 0, it provides the base case of the induction and determines all other components.
Remark A.1. The algorithm of this proof can be viewed as collapsing the local maxima till we end up at the last component, which has k local maxima with height 1. Since a Dyck path cannot fall below the x-axis we cannot collapse a local maximum with a height 1. Thus, the algorithm never uses (A.1) at j if the height a j is one. This is an important remark for the proofs that follow.
Remark A.2. The Dyck path L 0 corresponds to the link pattern that connects point i with 2k − i + 1. The same arguments used in this proof can be found in [10] where they prove the unique solution for the model with reflecting boundaries. k +2 * in L 2k and L 2k+1 , respectively. A little arch in a punctured link patterns is an arch connecting points j, j + 1 that does not contain the puncture. For example L ∩ only has one little arch, it connects points 2k + 1 to 1.
We show that the solution g (n) (z) ∈ Sol n (C[L n ]; q, c n ) is determined by its base component g where e i L ′ ∼ L means that L is obtained from e i L ′ by removing the loops in e i L ′ (there is in fact at most one loop), see (4.1). The coefficient γ if e i L ′ has a non null-homotopic loop, 1 otherwise.
We treat the even and odd case separately.
denote the set of punctured link patterns in L 2k such that the puncture could be connected to a point on the boundary between points j and j + 1 (modulo 2k) without crossing a line. Then, L 2k = 2k j=1 LP ( * ,j) 2k (not necessarily disjoint) and ρ : LP and if we define L 2k := ρ k · L ∩ then L 2k ∈ LP ( * ,2k) 2k . Define a bijection from LP ( * ,2k) 2k to LP 2k by simply removing the puncture. This mapping preserves the action of T L f 2k . Furthermore, it maps L 2k ∈ LP ( * ,2k) 2k to L 0 ∈ LP 2k . To prove that g  to LP 2k and the proof on LP 2k we have g There is one key subtlety that we have ignored, which we point out and address. There is a difference between equations (A.1) and (A.2); in the latter equation the pre-images are in L 2k and not just LP 2k . So when determining all the components for L ∈ LP ( * ,2k) 2k we must check that all the pre-images L ′ are also in LP to LP 2k by simply removing the defect line, puncture and boundary point 2k + 1. This mapping preserves the action of T L f 2k . Furthermore, it maps L 2k+1 ∈ LP ( * ,2k+1) 2k+1 to L 0 ∈ LP 2k . Now to prove that g to LP 2k and the proof on LP 2k we have g is the link pattern with either point i or i + 1 connected to the puncture and the other to the point 2k + 1.
A.3. The restricted module V νn n+1 . Here we present the proof to Lemma 4.3. Let g (n) (z) ∈ Sol n (V νn n+1 ; q, c n ). The qKZ equations associated with the representation V νn n+1 written componentwise are, L ′ (z 2 , . . . , z n , q −1 z 1 ) + t ρ·L (z) (A.5) for 1 ≤ j < n. It is important to note that the link patterns are in L n+1 but the first equation is only for 1 ≤ j < n; there is one equation less than in the previous cases. Note furthermore that (A.5) follows from the fact that I n (ρ) = ρ(t − 1 4 e n + t 1 4 ). The proof for the even and odd case are treated separately.
A.3.1. The case n = 2k. Note that for the case n = 2k the link patterns are in L 2k+1 . Recall from subsection A.2 the definitions for L ∩ , L 2k+1 ∈ L 2k+1 and LP ( * ,j) 2k+1 . We show the solution g (2k) (z) is determined by its base component g to LP 2k decreases the size of the link patterns by one.
The last step is to use equation (A.5). However, this is not as simple as subsection A.2 because equation (A.5) has an extra term on the left-hand side. It is a sum over pre-images for the action of e n . We will refer to it as the pre-image sum. Consider equation (A.5) for L ∈ LP ( * ,2k) 2k+1 . Since L has the defect line connected to point 2k there are no pre-images L ′ such that e 2k · L ′ ∼ L. Therefore the pre-image-sum of (A.5) does not give a contribution and g 2k+1 . This completes the induction and the base case is i = 1 which was discussed in the previous paragraph.
A.3.2. The case n = 2k − 1 (k ≥ 1). Note that for the case n = 2k − 1 the link patterns are in L 2k Recall from subsection A.2 the definitions for L ∩ , L 2k ∈ L 2k and LP ( * ,j) 2k . We show the solution g (2k−1) (z) is determined by its base component g (2k−1) L∩ (z) in three steps. The first step is identical to the case n = 2k. The link pattern L ∩ does not have a little arch connecting (2k − 1, 2k) and neither do the link patterns ρ j · L ∩ for 1 ≤ j ≤ k. So if g  . Since L does not have a little arch connecting points (2k − 1, 2k), there is no pre-image L ′ and hence the pre-image sum does not give a contribution. Therefore, we have g for j = 2k − 2, 2k − 1, 2k. Therefore, the pre-image sum gives no contribution and g (see Figure 4). Therefore, the pre-image sum is equivalently zero and since g . This completes the inductive step. The base case is i = 1 which was discussed in the beginning of this paragraph.