Quasisymmetric expansion of Hall-Littlewood symmetric functions

TL;DR

This paper introduces a quasisymmetric expansion of Hall-Littlewood symmetric functions using $q$-fundamental functions, bridging fundamental and peak functions and connecting to Schur functions.

Contribution

The authors develop a new quasisymmetric expansion of Hall-Littlewood functions via $q$-fundamental functions, unifying previous frameworks and extending their applicability.

Findings

$q$-fundamental functions interpolate between fundamental and peak functions.

Hall-Littlewood $S$-symmetric functions are expanded quasisymmetrically with parameter $t=-q$.

Provides a new perspective linking symmetric and quasisymmetric functions.

Abstract

In our previous works we introduced a $q$-deformation of the generating functions for enriched $P$-partitions. We call the evaluation of this generating functions on labelled chains, the $q$-fundamental quasisymmetric functions. These functions interpolate between Gessel's fundamental ($q=0$) and Stembridge's peak ($q=1$) functions, the natural quasisymmetric expansions of Schur and Schur's $Q$-symmetric functions. In this paper, we show that our $q$-fundamental functions provide a quasisymmetric expansion of Hall-Littlewood $S$-symmetric functions with parameter $t=-q$.