A q-analog of Schur's Q-functions

TL;DR

This paper introduces a new family of symmetric functions as q-analogs of Schur's Q-functions, revealing parallels with Kostka-Foulkes polynomials and expanding the algebraic framework of symmetric functions.

Contribution

It constructs a q-analog family of Hall-Littlewood functions within the Q-function algebra, detailing their basis change coefficients and properties.

Findings

Coefficients are q-analogs of marked shifted tableaux counts

Coefficients share properties with Kostka-Foulkes polynomials

Provides new algebraic tools for symmetric functions

Abstract

We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux. These coefficients exhibit many parallel properties to the Kostka-Foulkes polynomials.