# 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.

## Key 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 $n$.

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Source: https://tomesphere.com/paper/1903.08677