# Chromatic quasisymmetric functions for signed graphs

## TL;DR

This paper extends the concept of chromatic quasisymmetric functions to signed graphs, introducing a new invariant and algebra that deepen the algebraic combinatorics understanding of signed graph colorings.

## Contribution

It defines the signed chromatic quasisymmetric invariant and the algebra of signed quasisymmetric functions, expanding the framework to signed graphs and their associated hyperplane arrangements.

## Key findings

- Introduced the signed chromatic quasisymmetric invariant.
- Established structural properties of the invariant.
- Defined and studied the algebra SQSym.

## Abstract

In 1995, Stanley introduced the chromatic symmetric function of a graph, which specializes to its chromatic polynomial, and which has been the focus of intense research. In 2017, Shareshian, Wachs, and Ellzey defined a refinement of this function for a directed graph, that appears to be in $QSym$, the algebra of quasisymmetric functions, which is of great interest in algebraic combinatorics. Our goal is to extend this work to signed graphs, taking into account the perspective of the hyperplane arrangement associated with a signed graph, developed by Zaslavsky. We introduce the signed chromatic quasisymmetric invariant, and obtain structural properties. As a consequence, we define and study $SQSym$, the algebra of signed quasisymmetric functions.