Non-Symmetric Hall-Littlewood Polynomials

TL;DR

This paper introduces two new polynomial bases related to Hall-Littlewood polynomials using Hecke algebra actions, revealing their connections to classical Key polynomials and establishing an adjoint scalar product.

Contribution

It defines two polynomial bases via Hecke algebra actions, generalizing Hall-Littlewood and Key polynomials, and introduces an adjoint scalar product for these bases.

Findings

The bases generalize Hall-Littlewood polynomials.

Specialization at q=0 yields classical Key polynomials.

An adjoint scalar product for the bases is established.

Abstract

Using the action of the Yang-Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall-Littlewood polynomials are a subfamily of one of them. For q=0, these bases specialize into the two families of classical Key polynomials (i.e. Demazure characters for type A). We give a scalar product for which the two bases are adjoint of each other.