Representation theoretic interpretation and interpolation properties of inhomogeneous spin q-Whittaker polynomials

Abstract

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing \(t=0\) Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of \(U'_q({\widehat{\mathfrak {sl}}}_2)\) -modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials.