# Chromatic symmetric functions in noncommuting variables revisited

## TL;DR

This paper advances the understanding of chromatic symmetric functions in noncommuting variables by classifying positivity conditions, establishing algebraic properties, and exploring their bases and realizability.

## Contribution

It extends the study of chromatic symmetric functions from commuting to noncommuting variables, including classification, multiplicativity, and basis construction.

## Key findings

- Classified when $Y_G$ is a positive combination of elementary and Schur functions.
- Proved multiplicativity and $k$-deletion property for $Y_G$.
- Established new bases for the algebra of symmetric functions in noncommuting variables.

## Abstract

In 1995 Stanley introduced a generalization of the chromatic polynomial of a graph $G$, called the chromatic symmetric function, $X_G$, which was generalized to noncommuting variables, $Y_G$, by Gebhard-Sagan in 2001. Recently there has been a renaissance in the study of $X_G$, in particular in classifying when $X_G$ is a positive linear combination of elementary symmetric or Schur functions. We extend this study from $X_G$ to $Y_G$, including establishing the multiplicativity of $Y_G$, and showing $Y_G$ satisfies the $k$-deletion property. Moreover, we completely classify when $Y_G$ is a positive linear combination of elementary symmetric functions in noncommuting variables, and similarly for Schur functions in noncommuting variables, in the sense of Bergeron-Hohlweg-Rosas-Zabrocki. We further establish the natural multiplicative generalization of the fundamental theorem of symmetric functions, now in noncommuting variables, and obtain numerous new bases for this algebra whose generators are chromatic symmetric functions in noncommuting variables. Finally, we show that of all known symmetric functions in noncommuting variables, only all elementary and specified Schur ones can be realized as $Y_G$ for some $G$.