q-symmetric functions and q-quasisymmetric functions

TL;DR

This paper develops q-analogues of algebraic structures related to symmetric functions, introduces odd and q-quasisymmetric functions, and explores their combinatorial and algebraic properties, including dual graphs and rules.

Contribution

It constructs q-analogues of Poirier-Reutenauer algebras and introduces odd quasisymmetric Schur functions, expanding the framework of q- and odd symmetric functions.

Findings

Construction of q-analogues of Poirier-Reutenauer algebras

Realization of odd Schur functions and Littlewood-Richardson rule

Development of odd quasisymmetric Schur functions and associated q-dual graphs

Abstract

In this paper, we construct the q-analogue of Poirier-Reutenauer algebras, related deeply with other q-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined in \cite{EK}, and naturally obtain the odd Littlewood-Richardson rule concerned in \cite{Ell}. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration in \cite{HLMW1}. All the q-Hopf algebras we discuss here provide the corresponding q-dual graded graphs in the sense of \cite{BLL}.