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

$$t=0$$

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

$$U'_q({\widehat{\mathfrak {sl}}}_2)$$

U q ′

(

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.