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 \(\mathrm{GL}_n\)-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 \({\mathcal {H}}_n\rightarrow {\mathcal {H}}_{n+1}\), in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper, we consider qKZ towers \((f^{(n)})_{n\ge 0}\) 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 the 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 \(({\mathcal {H}}_n)_{n\ge 0}\). We associate with 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 specialized dual non-symmetric Macdonald polynomials with specialized 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.