Koornwinder polynomials and the XXZ spin chain
Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin- chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley-Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZ spin chain, in the form of matchmaker representations they relate to Temperley-Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik-Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.
Key words and phrases:
2000 Mathematics Subject Classification:
Dedicated to the 80th birthday of Dick Askey.
1.1. Nonsymmetric Koornwinder polynomials
The four-parameter family of Askey-Wilson polynomials, introduced in 1985 in the famous monograph , is the most general class of classical -orthogonal polynomials in one variable. Important generalizations have appeared since: the associated Askey-Wilson polynomials , Rahman’s  biorthogonal rational functions, nonsymmetric Askey-Wilson polynomials [50, 60], Askey-Wilson functions , elliptic analogs of Askey-Wilson polynomials [64, 56], etc.
There exist two completely different multivariable extensions of the Askey-Wilson polynomials. The first is due to Gasper and Rahman . In this case the multivariable Askey-Wilson polynomials admit explicit multivariate basic hypergeometric series expansions, allowing the corresponding theory to be developed by classical methods. See  for the corresponding multivariable generalization of the Askey-Wilson function.
The second type of multivariable generalization of the Askey-Wilson polynomials is part of the Macdonald-Cherednik [43, 10, 72] theory on root system analogs of continuous -ultraspherical, continuous -Jacobi and Askey-Wilson polynomials. In this case the multivariable extension of the Askey-Wilson polynomial, the nonsymmetric Askey-Wilson polynomial and the Askey-Wilson function are the Koornwinder polynomial , the nonsymmetric Koornwinder polynomial  and the basic hypergeometric function [68, 70], respectively. These multivariable extensions depend on five parameters and arise in harmonic analysis on (quantum) symmetric spaces [49, 41], representation theory of affine Hecke algebras [10, 60] and in quantum relativistic integrable one-dimensional many-body systems of Calogero-Moser type [59, 13]. Multivariable extensions of Rahman’s  biorthogonal rational functions and of the elliptic analogs of the Askey-Wilson functions are due to Rains .
1.2. Heisenberg spin chains
Spin models originate in the statistical mechanical study [5, 6, 27] of magnetism. The Heisenberg spin chain is a one-dimensional quantum model of magnetism with a quantum spin particle residing at each site of a one-dimensional lattice. The interaction of the quantum spin particles is given by nearest neighbour spin-spin interaction. Assuming that the number of sites is finite, say , boundary conditions need to be imposed to determine how the quantum spin particles at the boundary sites and are treated in the model. The most investigated choice is to impose periodic (or closed) boundary conditions, in which case we assume that the sites and are also adjacent (the one-dimensional lattice is put on a circle). In this paper we focus on so-called reflecting (or open) boundary conditions, in which case the quantum spin particles at the boundary sites and are interacting with reflecting boundaries on each side.
Then is the linear operator on given by
where the subindices indicate the tensor leg of on which the Pauli matrix is acting. Besides the three coupling constants and there are three parameters and at the left reflecting boundary and three parameters and at the right reflecting boundary of the one-dimensional lattice. In general, the three coupling constants are distinct, in which case corresponds to the XYZ spin- model with general reflecting boundary conditions; in its full generality it was first obtained in . In this paper we consider the special case that , in which case the spin chain is the XXZ spin- model with general reflecting boundary conditions. We denote the resulting quantum Hamiltonian by . Up to rescaling, thus has seven free parameters; also see [74, 47].
The XXZ spin- chain with generic reflecting boundary conditions is quantum integrable, in the sense that the quantum Hamiltonian is part of a large commuting set of linear operators on encoded by a transfer operator (see ). For the transfer operator is produced from the standard XXZ spin- solution of the quantum Yang-Baxter equation and three-parameter solutions of the associated left and right reflection equations, see  and Subsection 4.4. It opens the way to study the spectrum and eigenfunctions of using Sklyanin’s  generalization of the algebraic Bethe ansatz to quantum integrable models with reflecting boundary conditions.
1.3. Representation theory
A representation theoretic context for the five parameter family of Koornwinder polynomials is provided by Letzter’s  notion of quantum symmetric pairs. The quantum symmetric pairs pertinent to Koornwinder polynomials are given by a family of coideal subalgebras of the quantized universal enveloping algebra of naturally associated to quantum complex Grassmannians (see [49, 51]). For the XXZ spin- spin chain with general reflecting boundary conditions, a representation theoretic context is provided by coideal subalgebras of the quantum affine algebra of known as -Onsager algebras (see ). -Onsager algebras are examples of quantum affine symmetric pairs .
The five-parameter double affine Hecke algebras of type  provides another representation theoretic context for Koornwinder polynomials. On the other hand, various special cases of the XXZ spin- chain with general reflecting boundary conditions have been related to the three-parameter affine Hecke algebra of type , which is a subalgebra of the double affine Hecke algebra (see  and references therein).
It is one of the purposes of this paper to show that the double affine Hecke algebra governs the whole seven-parameter family of XXZ spin- chains with reflecting boundary conditions. The double affine Hecke algebra gives rise to a Baxterization procedure for affine Hecke algebra representations, producing for a given representation a two-parameter family of solutions of quantum Yang-Baxter and reflection equations (see Subsection 4.1). In the case of spin representations, which form a two-parameter family of representations of the affine Hecke algebra of type on the state space [23, 21], we end up with solutions depending on seven parameters: three affine Hecke algebra parameters denoted by , two Baxterization parameters denoted by , and two spin representation parameters denoted by . The resulting transfer operator reproduces as the associated quantum Hamiltonian under an appropriate parameter correspondence. The parameters are the five parameters of the Koornwinder polynomials.
The spin representations factor through the two-boundary Temperley-Lieb algebra . The two-boundary Temperley-Lieb algebra has a natural two-parameter family of representations on the formal vector spaces spanned by two-boundary non-crossing perfect matchings (see Definition 3.6), in which the generators of the algebra act by ”matchmakers”, see, e.g., [20, 21] and Subsection 3.3. It is known  that generically the resulting matchmaker representations are isomorphic to the spin representations under a suitable parameter correspondence. We provide a new proof for this in Subsection 3.3 by constructing an explicit intertwiner. The Baxterization of the matchmaker representation leads to a transfer operator acting on the formal vector space of two-boundary non-crossing perfect matchings. The associated quantum Hamiltonian corresponds to the Temperley-Lieb loop model (also known as dense loop model) with general open boundary conditions (see Subsection 4.4); special cases are discussed in, e.g., [44, 20, 21]. It leads to the conclusion that generically the XXZ spin- chain and the Temperley-Lieb loop model for general reflecting boundary conditions are equivalent under a suitable parameter correspondence.
1.4. Weyl group invariant solutions of the reflection quantum KZ equations
The quantum Knizhnik-Zamolodchikov (KZ) equations are a consistent system of first order linear -difference equations, which were first studied by Smirnov in relation to integrable quantum field theories . They were related to representation theory of quantum affine algebras by Frenkel and Reshetikhin . Their solutions entail correlation functions for XXZ spin chains with quasi-periodic boundary conditions, see, e.g., . More recently, links to algebraic geometry have been exposed . Generalizations of these equations for arbitrary root systems were constructed by Cherednik [8, 9]. For type A, the aforementioned Smirnov-Frenkel-Reshetikhin quantum KZ equations are recovered; reflection quantum KZ equations are the equations in Cherednik’s framework associated to the other classical types [8, §5]. In the context of the spin- Heisenberg chain, the reflection quantum KZ equations were first derived in .
A choice of solutions of the quantum Yang-Baxter equation and the associated reflection equations gives rise to reflection quantum KZ equations . In case the solutions are obtained from the Baxterization of the spin representations we obtain reflection quantum KZ equations depending on seven parameters. These are the reflection quantum KZ equations under investigation in the last section of the paper (Section 5). Their solutions are expected to give rise to correlation functions for the spin chain governed by . The close link to the spin chain is apparent from the fact that the transport operators of the reflection quantum KZ equations at are higher quantum Hamiltonians for the spin chain (see Subsection 4.3).
For special choices of reflecting boundary conditions methods have been developed to construct explicit solutions of the reflection quantum KZ equations, e.g. in [30, 31, 75, 12, 11, 77, 52, 58]. In Section 5 we construct solutions using nonsymmetric Koornwinder polynomials, following the ideas from [34, 70].
The solution space of the reflection quantum KZ equations associated to has a natural action of the Weyl group of type , which is the hyperoctahedral group . In Section 5 we show that generically, the space of Weyl group invariant solutions is in bijective correspondence to a suitable space of eigenfunctions of the Cherednik-Noumi  -operators. The -operators are commuting scalar-valued -difference-reflection operators generalizing Dunkl operators. The link is provided by the nonsymmetric Cherednik-Matsuo correspondence . This correspondence can be applied in the present context, because the spin representations are examples of principal series representations (see Subsection 3.2).
The nonsymmetric basic hypergeometric function of Koornwinder type [68, 70] provides distinguished examples of eigenfunctions of the -operators. We show that it produces a nontrivial -invariant solution of the reflection quantum KZ equations for generic values of the parameters. It is exactly known under which specialization of the spectral parameters the nonsymmetric basic hypergeometric function reduces to a nonsymmetric Koornwinder polynomial. It follows that if the parameters satisfy one of the following two conditions,
then nontrivial -invariant Laurent polynomial solutions of the reflection quantum KZ equations exist (see Theorem 5.10). Note that these conditions do not depend on the two Baxterization parameters . -invariant Laurent polynomial solutions are expected to play an important role in generalizations of Razumov-Stroganov conjectures , cf., e.g., [24, 76, 35].
It is an open problem how the solutions of the reflection quantum KZ equations in terms of nonsymmetric basic hypergeometric functions and nonsymmetric Koornwinder polynomials are related to other constructions of explicit solutions [30, 31, 75, 12, 11, 77, 52, 58]. Wall-crossing formulas for the reflection quantum KZ equations associated to are in reach by [69, 71]. They are expected to lead to elliptic solutions of quantum dynamical Yang-Baxter equations and associated dynamical reflection equations governing the integrability of elliptic solid-on-solid models with reflecting ends, cf.  for such models with diagonal reflecting ends. Pursuing the techniques of the present paper at the elliptic level, either for the XYZ spin chain or elliptic solid-on-solid models, is expected to lead to connections to elliptic hypergeometric functions [55, 64, 65, 66]. Only some first steps have been made here, see, e.g., [78, 73].
We are grateful to P. Baseilhac, J. de Gier, B. Nienhuis, N. Reshetikhin and P. Zinn-Justin for stimulating discussions. The work of B.V. was supported by an NWO Free Competition grant (”Double affine Hecke algebras, Integrable Models and Enumerative Combinatorics”).
Throughout the paper . Linear operators of are represented as -matrices with respect to a fixed ordered basis of , and linear operators of are represented as -matrices with respect to the ordered basis of .
2. Groups and algebras of type \excepttoc\fortoc
2.1. Affine braid group
The affine braid group of type is the group with generators subject to the braid relations
It can be topologically realized as follows (see the type case in [2, §4]). Consider the line segments and in . Let be the group of -braids in the strip , with the -braids attached to the floor at () and to the ceiling at (). The group operation is putting the -braid on top of and shrinking the height by isotopies. The group isomorphism is induced by the identification
where the right hand sides are the braid diagram projections on the -plane. For , it is easy to check topologically that the elements
of the affine braid group pairwise commute.
2.2. The affine Weyl group
The affine Weyl group of type is the quotient of by the relations (). The generators descend to group generators of , which we denote by . The affine Weyl group is a Coxeter group with Coxeter generators . Write for the commuting elements in corresponding to under the canonical surjection , so that
The affine Weyl group acts faithfully on by affine linear transformations via
for . Note that the act on as orthogonal reflections in affine hyperplanes. We sometimes call the simple reflections of . The subgroup of generated by , acting by permutations and sign changes of the coordinates, is isomorphic to with the symmetric group in letters (which, in turn, is isomorphic to the subgroup generated by ). Note furthermore that
hence the abelian subgroup of generated by is isomorphic to . We write for . We have , with acting on by permutations and sign changes of the coordinates.
2.3. Affine Hecke algebra
Fix with . We write for the value ().
The affine Hecke algebra of type is the quotient of the group algebra of the affine braid group by the two-sided ideal generated by the elements for . The generators descend to algebraic generators of , which we denote by .
The quadratic relation in implies that is invertible in with inverse .
For the affine Hecke algebra is the group algebra of the affine Weyl group of type .
Let be the element in corresponding to under the canonical surjection . Then
The pairwise commute in . They are sometimes called Murphy elements (cf., e.g., [21, Def. 2.8]). In the context of representation theory of affine Hecke algebras, they naturally arise in the Bernstein-Zelevinsky presentation of the affine Hecke algebra, see .
The elements and () generate .
2.4. The two-boundary Temperley-Lieb algebra
In analogy to the setup for the affine Hecke algebra, fix with . The two-boundary Temperley-Lieb algebra TL , also known as the open Temperley-Lieb algebra , is the unital associative algebra over with generators satisfying
To generic affine Hecke algebra parameters we associate two-boundary Temperley-Lieb parameters by
for and . In the remainder of the paper we always assume that the affine Hecke algebra parameters and the two-boundary Temperley-Lieb algebra parameters are matched in this way. Then there exists a unique surjective algebra homomorphism such that
3.1. Spin representation
In the following lemma we give a two-parameter family of representations of the two-boundary Temperley-Lieb algebra on the state space of the Heisenberg XXZ spin- chain. It includes the one-parameter family of representations that has been intensively studied in the physics literature (see, e.g., [21, §2.3] and references therein).
We use the standard tensor leg notations for linear operators on . Note also the convention on the matrix notation for linear operators on as given at the end of the introduction.
Let the free parameters of the two-boundary Temperley-Lieb algebra be given in terms of the generic affine Hecke algebra parameters by (2.5). Let . There exists a unique algebra homomorphism
This is a straightforward verification. ∎
We lift to a representation
of the affine Hecke algebra via the surjection , so . Then
and the flip operator . The braid relations of imply that is a solution of the quantum Yang-Baxter equation and that and are solutions to associated reflection equations (all equations without spectral parameter). The solution is the well-known solution of the quantum Yang-Baxter equation arising from the universal -matrix of the quantized universal enveloping algebra acting on , with viewed as the vector representation of .
The representations and are called spin representations of and , respectively.
3.2. Principal series representation
One of the main aims of this paper is to relate solutions of reflection quantum Knizhnik-Zamolodchikov equations for Heisenberg XXZ spin- chains with boundaries to Macdonald-Koornwinder type functions. Such a connection is known in the context of so-called principal series representations of affine Hecke algebras; see Subsection 5.1 and references therein. Principal series representations are a natural class of finite dimensional representations of the affine Hecke algebra, which we define now first. In the second part of this subsection we show that the spin representation is isomorphic to a principal series representation.
Set . Note that acts on by permutations and inversions of the coordinates.
Let be the subalgebra generated by () and ().
Let be the elements satisfying if and satisfying if .
Let . There exists a unique algebra map satisfying
The proof is a straightforward adjustment of the proof of [69, Lem. 2.5(i)]. ∎
We write for viewed as -module with representation map .
Let and . The associated principal series module is the induced -module . We write for the corresponding representation map.
Concretely, with the -action given by
The principal series module is finite dimensional. In fact,
with the subgroup of generated by the (). The -orbit of in is called the central character of .
Let . We have
Note that for , hence is well defined.
Observe that is a cyclic -representation with cyclic vector . Note furthermore that
In addition, for ,
Hence for . We thus have a surjective linear map mapping to for all , which intertwines the -action on with the -action on . A dimension count shows that it is an isomorphism. ∎
3.3. Matchmaker representation
In this subsection we revisit some of the key results from  and reestablish them by different methods.
The diagrammatic realization of (boundary) Temperley-Lieb algebras (see, e.g., [20, 23, 21]) gives rise to natural examples of Temperley-Lieb algebra actions on linear combinations of non-crossing matchings (or equivalently, link patterns) in which the Temperley-Lieb algebra generators act as matchmakers. We give a two-parameter family of such matchmaker representations of . We reestablish the result from  that this family is generically isomorphic to the family of spin representations of by constructing an explicit intertwiner involving a subtle combinatorial expression given in terms of the various orientations of the pertinent non-crossing matchings. The origin of such explicit intertwiners traces back to the explicit link between loop models and the XXZ spin chain from [44, §8]. In the quasi-periodic context, related to the affine Hecke algebra of type and the affine Temperley-Lieb algebra, such intertwiners were considered in [77, §3.2] and [46, Prop. 3.2].
Let be the set of unordered pairs () of and denote by the power set of . If then is said to be matched to if .
is called a two-boundary non-crossing perfect matching if the following conditions hold:
Each is matched to exactly one .
There are no pairs with .
We write for the set of two-boundary non-crossing perfect matchings of .
Note that the definition of two-boundary non-crossing perfect matching allows and to have multiple matchings; these boundary points may also be unmatched. We can give a diagrammatic representation of by connecting and by an arc in the upper half plane for all . If then this can be done in such a way that the arcs do not intersect except possibly at the boundary endpoints and . For and we write for the unique element such that . Next we set
The map defined by , is a bijection (cf., e.g., ). Consequently .
Let be the formal vector space over with basis the two-boundary non-crossing perfect matchings . We define now an action of the two-boundary Temperley-Lieb algebra on , depending on two parameters , in which the generators act by so-called matchmakers. The idea is as follows. If and then the action of on replaces all pairs containing and/or and adds pairs and . When this does not produce a two-boundary non-crossing perfect matching, either because and are both boundary points or because , then only the pair is matched and the omission of the second pair is accounted for by a suitable multiplicative constant (in this case, the deleted pair is either an arc from to for some or an arc from to ). Thus the action of on a two-boundary non-crossing perfect matching always has the effect that it matches up and . A similar matchmaker interpretation is given for the boundary operators and , which match up with and with , respectively.
We now give the precise formulation of the resulting matchmaker representation of . For and write to be the element in obtained from by removing any pairs containing and/or . Similarly, for we write to be the element in obtained from by removing any pair containing , respectively. For write if is even and if is odd.
Fix . There exists a unique algebra homomorphism
such that for and is given by
for , and
It is sufficient to verify that the defining relations (2.2)-(2.4) of are satisfied by the matchmakers . This can be done by a straightforward case-by-case analysis, relying in part on the fact that if for and , then . An instructive example is the case for such that . Then
where for . Since the parities of and must be different, hence . Hence , as requested. ∎
See, e.g., [20, 21] for a discussion of the diagrammatic representation of the action of Temperley-Lieb algebras on matchings. It is very helpful for direct computations. An example of such a diagrammatic computation for our representation of on () is
where the left and right vertical line should be thought of as the boundary point and , respectively, and where we have represented by its image under the bijection .
Note that the spin representation is isomorphic to if in view of Proposition 3.5. We now show that the -dimensional -representations and are isomorphic for generic parameters if