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

TL;DR

This paper constructs solutions to qKZ equations forming a tower compatible with affine Hecke algebra structures, applying them to dense loop models and connecting to Macdonald polynomials and ground states of loop models.

Contribution

It introduces a new framework for qKZ towers compatible with algebraic structures and relates solutions to loop models using specialized Macdonald polynomials.

Findings

Constructed qKZ towers from specialized Macdonald polynomials.

Connected solutions to the dense O(1) loop model ground state.

Demonstrated compatibility with affine Hecke algebra tower structure.

Abstract

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 $n$.