Double posets and the antipode of QSym

TL;DR

The paper introduces a new formula for the antipode of quasisymmetric functions associated with double posets, generalizing previous results and including cases with group actions.

Contribution

It provides a new, self-contained proof of the antipode formula for double poset quasisymmetric functions, extending previous work to broader settings.

Findings

The antipode formula applies when the second order of the double poset is total.

The proof is new and self-contained, differing from prior approaches.

The results include cases with group actions on double posets.

Abstract

A quasisymmetric function is assigned to every double poset (that is, every finite set endowed with two partial orders) and any weight function on its ground set. This generalizes well-known objects such as monomial and fundamental quasisymmetric functions, (skew) Schur functions, dual immaculate functions, and quasisymmetric $\left(P, \omega\right)$-partition enumerators. We prove a formula for the antipode of this function that holds under certain conditions (which are satisfied when the second order of the double poset is total, but also in some other cases); this restates (in a way that to us seems more natural) a result by Malvenuto and Reutenauer, but our proof is new and self-contained. We generalize it further to an even more comprehensive setting, where a group acts on the double poset by automorphisms.