The Chromatic Quasisymmetric Class Function of a Digraph

TL;DR

This paper introduces a new quasisymmetric class function for group actions on digraphs and double posets, generalizing previous functions and establishing representation-theoretic and combinatorial properties.

Contribution

It generalizes the chromatic quasisymmetric function to group actions on digraphs and posets, and proves new representation-theoretic and reciprocity results.

Findings

Proves $F$-positivity for the new class function

Establishes combinatorial reciprocity theorems

Introduces orbital quasisymmetric functions

Abstract

We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the latter is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including $F$-positivity, and combinatorial reciprocity theorems. We also deduce results for orbital quasisymmetric functions. We also study a generalization of the notion of strongly flawless sequences.