Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Abstract

Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for Vn-valued meromorphic functions on a complex n-torus, with Vn a module over the GLn-type extended affine Hecke algebra Hn. The family (Hn) n≥0 of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms Hn → H n+1 in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper we consider qKZ towers (f (n) ) n≥0 of solutions, which consist of twisted-symmetric polynomial solutions f (n) (n ≥ 0) of the qKZ equations that are compatible with the tower structure on (Hn) n≥0 . The compatibility is encoded by so-called braid recursion relations: f (n+1) (z 1 , . . . , zn, 0) is required to coincide up to a quasiconstant factor with the push-forward of f (n) (z 1 , . . . , zn) by an intertwiner µn : Vn → V n+1 of Hn-modules, where V n+1 is considered as an Hn-module through the tower structure on (Hn) n≥0 .We associate to the dense loop model on the half-infinite cylinder with nonzero loop weights a qKZ tower (f (n) ) n≥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. J for J ⊆ {1, . . . , ⌊n/2⌋}\{i}. These equalities follow from the following diagrammatic calculations, in which we omit all paths that are not involved in the computation. The first diagrammatic computation is for Y 2i D 2k J in V 2k , the second for Y 2i D 2k+1 J

The skein category of the annulus

“…In the following proposition we explicitly compute I n on the algebraic generators of TL n . 4 ) and I 1 (ρ −1 ) = (t 4 ), 4 ),…”

“…We construct examples that are relevant for understanding the dependence of dense loop models and Heisenberg XXZ spin- 1 2 chains on their system size (cf. [16,6,4]). We first introduce some notations.…”

“…Motivated by applications to integrable models in statistical physics [16,6,4], in particular to the dense loop model and the Heisenberg XXZ spin− 1 2 chain, we construct in this section a family of towers of extended affine Temperley-Lieb algebra modules acting on spaces of link patterns on the punctured disc. We use the skein categorical context to build the tower.…”

“…In our future work [4] we use the link pattern tower to construct a tower of solutions to quantum Knizhnik-Zamolodchikov (qKZ) equations. The tower consists of V n -valued solutions of qKZ equations (n ∈ Z ≥0 ) which are compatible with respect to the connecting maps φ n .…”

“…Acknowledgments. The authors would like to thank Bernard Nienhuis as motivation for this paper came from the joint work [4]. We thank the two referees of the paper for their comments which led to substantial improvements of the paper.…”