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 $({\mathcal{H}}_n{)}_{n\ge 0}$ of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms ${\mathcal{H}}_n\to {\mathcal{H}}_{n+1}$ the Hecke algebra descends of arc insertion at the affine braid group level. In this paper we consider qKZ towers $(f^{(n)}{)}_{n\ge 0}$ of solutions, which consist of twisted-symmetric polynomial solutions $f^{(n)}$ ($n\ge 0$) of the qKZ equations that are compatible with the tower structure on $({\mathcal{H}}_n{)}_{n\ge 0}$. The compatibility is encoded by so-called braid recursion relations: $f^{(n+1)}(z_1,\dots ,z_n,0)$ is required to coincide up to a quasi-constant factor with the push-forward of $f^{(n)}(z_1,\dots ,z_n)$ by an intertwiner ${\mu }_n:V_n\to V_{n+1}$ of ${\mathcal{H}}_n$-modules, where $V_{n+1}$ is considered as an ${\mathcal{H}}_n$-module through the tower structure on $({\mathcal{H}}_n{)}_{n\ge 0}$. We associate to the dense loop model on the half-infinite cylinder with nonzero loop weights a qKZ tower $(f^{(n)}{)}_{n\ge 0}$ 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 $n$.