A Survey of $q$-Whittaker polynomials

TL;DR

This survey explores the properties, identities, and combinatorial aspects of $q$-Whittaker polynomials, highlighting their connections to Macdonald polynomials and symmetric group modules.

Contribution

It provides a comprehensive overview of $q$-Whittaker polynomials, including their derivation from Macdonald polynomials and combinatorial Pieri formulas.

Findings

Derived basic properties of $q$-Whittaker polynomials.

Established identities and their combinatorial interpretations.

Connected $q$-Whittaker polynomials to graded Frobenius characteristics.

Abstract

Exploiting the fact that the $q$-Whittaker polynomials arise as a specialization of the (modified) Macdonald polynomials, we derive some of their basic properties, and explore interesting identities that they satisfy. We also show how they arise as graded Frobenius characteristics of $\S_n$-modules, and give a combinatorial approach to associated Pieri formulas.