Inhomogeneous $q$-Whittaker polynomials II: ring theorem and positive specializations

TL;DR

This paper extends the theory of inhomogeneous $q$-Whittaker polynomials, showing they form a basis of a certain ring, describing positive specializations, and connecting them with Macdonald-positive specializations and probability distributions.

Contribution

It establishes that inhomogeneous $q$-Whittaker polynomials form a basis of an extended symmetric function ring and characterizes their positive specializations.

Findings

Inhomogeneous $q$-Whittaker polynomials form a basis of a certain ring.

Positive specializations are described and related to Macdonald-positive specializations.

Probability distributions are derived from positive specializations.

Abstract

We study inhomogeneous $q$-Whittaker polynomials which extend both $q$-Whittaker and stable Grothendieck polynomials. We prove that inhomogeneous $q$-Whittaker polynomials (in countably many variables) form a basis of certain commutative ring extending the ring of symmetric functions to a subring of its completion. We then describe positive specializations of that ring and relate them with a subset of Macdonald-positive specializations of the ring of symmetric functions. We also show some related probability distributions obtained from positive specializations of inhomogeneous $q$-Whittaker polynomials.