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 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)_{nge 0} of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms {mathcal {H}}_nrightarrow {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)})_{nge 0} of solutions, which consist of twisted-symmetric polynomial solutions f^{(n)} (nge 0) of the qKZ equations that are compatible with the tower structure on ({mathcal {H}}_n)_{nge 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)_{nge 0}. We associate with the dense loop model on the half-infinite cylinder with nonzero loop weights, a qKZ tower (f^{(n)})_{nge 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.