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

TL;DR

This paper introduces a new vertex model-based approach to inhomogeneous spin q-Whittaker polynomials, establishing their properties, a Cauchy identity, and interpolation formulas, expanding understanding of their algebraic and combinatorial structure.

Contribution

It provides a novel vertex model construction using intertwining operators, proving a general Cauchy identity and deriving interpolation properties for these polynomials.

Findings

Established a vertex model representation for inhomogeneous spin q-Whittaker polynomials.

Proved a comprehensive Cauchy-type identity for these polynomials.

Characterized polynomials via vanishing conditions and derived interpolation formulas.

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 $U'_q(\hat{\mathfrak{sl}}_2)$-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.