# Hopf algebra structure of symmetric and quasisymmetric functions in superspace

## TL;DR

This paper establishes the Hopf algebra structures of symmetric and quasisymmetric functions in superspace, providing explicit formulas and dualities, thus advancing the algebraic understanding of these functions.

## Contribution

It introduces the Hopf algebra sQSym of quasisymmetric functions in superspace and describes its algebraic operations explicitly, along with the dual ring sNSym.

## Key findings

- Ring of symmetric functions in superspace is a cocommutative, self-dual Hopf algebra.
- Explicit formulas for coproduct and antipode on various bases.
- Duality between sQSym and sNSym established.

## Abstract

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.