New degrees of freedom in integrable models with q-Hahn weights and their applications to symmetric functions and probability.

Abstract

We present three groups of results about integrable lattice models constructed from orthogonality weights of q-Hahn polynomials. First, we establish that the q-Hahn orthogonality weights appear as matrix coefficients in certain isomorphisms between tensor products of representations of quantum affine sl₂ algebra. This allows us to find new integrable degrees of freedom in q-Hahn models by constructing an integrable vertex model on a square lattice with weights coming not from an R-matrix, as usually the case, but from our isomorphisms.

Second, we use the partition function of our new vertex model to construct a generalization of t=0 Macdonald symmetric functions, which we call inhomogeneous spin q-Whittaker polynomials. Using integrability we are able to extend several classical properties of symmetric functions to our generalization, in particular, we prove analogues of the Cauchy and dual Cauchy identities. Moreover, we are able to characterize spin q-Whittaker polynomials by vanishing at certain points, which leads to a discovery of interpolation analogues of q-Whittaker and elementary symmetric polynomials.

Finally, we introduce a (colored) stochastic version of our vertex model and prove explicit integral expressions for q-deformed moments of the (colored) height functions of it. Following known techniques our stochastic model can be interpreted as a q-discretization of the Beta polymer model with three families of integrable parameters, and we are able to extend the known results about Tracy-Widom large-scale fluctuations to our generalization of this polymer model.