Koornwinder polynomials and the XXZ spin chain

Abstract

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-$\frac{1}{2}$ 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.

30 pages; corrected some minor mistakes immediately before and after Def. 3.6. appears in Journal of Approximation Theory (2014)

References in corpus (7)

• Essays on the theory of elliptic hypergeometric functions
• The two-boundary Temperley-Lieb algebra
• Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
• Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
• Exact finite size groundstate of the O(n=1) loop model with open boundaries
• Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices
• The polynomial representation of the double affine Hecke algebra of type $(C_n^{\vee },C_n)$ for specialized parameters

Cited by in corpus (11)

• Matrix product formula for Macdonald polynomials
• Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
• Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries
• Boundary transfer matrices and boundary quantum KZ equations
• Integral solutions to boundary quantum Knizhnik-Zamolodchikov equations
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• Connection problems for quantum affine KZ equations and integrable lattice models
• Tensor K-matrices for quantum symmetric pairs
• A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian
• Laurent polynomial solutions of the boundary quantum Knizhnik--Zamolodchikov equation
• The open XXZ chain at $\Delta =-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations