On the Hopf superalgebra of symmetric functions in superspace

TL;DR

This paper develops a superspace version of combinatorial Hopf algebras, establishing the Hopf superalgebra of symmetric functions in superspace as a terminal object and introducing a superspace analogue of chromatic symmetric functions.

Contribution

It introduces a superspace analogue of combinatorial Hopf algebras and characterizes the Hopf superalgebra of symmetric functions in superspace as a terminal object.

Findings

Hopf superalgebra of symmetric functions in superspace is a terminal object in its category.

Defined a superspace analogue of chromatic symmetric functions of graphs.

Established the structure of the Hopf superalgebra of quasi-symmetric functions in superspace.

Abstract

We introduce a superspace analogue of combinatorial Hopf algebras (Aguiar-Bergeron-Sottile, 2006), and show that the Hopf superalgebra of quasi-symmetric (resp. symmetric) functions in superspace (Fishel-Lapointe-Pinto, 2019) is a terminal object in the category of all (resp. cocommutative) combinatorial Hopf superalgebras. We also introduce a superspace analogue of chromatic symmetric functions of graphs (Stanley, 1995) using the chromatic Hopf superalgebra of two-colored graphs.