Hall-Littlewood vertex operators and generalized Kostka polynomials

TL;DR

This paper introduces a new family of vertex operators that generate symmetric functions linked to generalized Kostka polynomials, extending Jing's work and providing new algebraic relations and identities in equivariant K-theory.

Contribution

The paper presents a novel family of vertex operators that generalize Jing's operators for Hall-Littlewood functions, leading to new relations involving generalized Kostka polynomials.

Findings

Derived commutation relations for the new vertex operators

Established identities in (GL(n) x C^*)-equivariant K-theory

Connected symmetric functions to generalized Kostka polynomials

Abstract

A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka-Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n) x C^*)-equivariant K-theory of the nullcone.