On the Hopf algebra of noncommutative symmetric functions in superspace

TL;DR

This paper explores the structure of the Hopf algebra of noncommutative symmetric functions in superspace, introducing new bases, formulas, and a tree realization, advancing the algebraic understanding of superspace functions.

Contribution

It introduces primitive elements, extends elementary and power sum functions, and defines noncommutative Ribbon Schur functions in superspace with explicit product and coproduct formulas.

Findings

Defined noncommutative Ribbon Schur functions in superspace.

Provided explicit formulas for products and coproducts of these functions.

Established a Hopf algebra of trees realization for sNSym.

Abstract

We study in detail the Hopf algebra of noncommutative symmetric functions in superspace sNSym, introduced by Fishel, Lapointe and Pinto. We introduce a family of primitive elements of sNSym and extend the noncommutative elementary and power sum functions to superspace. Then, we give formulas relating these families of functions. Also, we introduce noncommutative Ribbon Schur functions in superspace and provide a explicit formula for their product. We show that the dual basis of these function is given by a family of the so--called fundamental quasisymmetric functions in superspace. This allows us to obtain a explicit formula for the coproduct of fundamental quasisymmetric functions in superspace. Additionally, by projecting the noncommutative Ribbon Schur functions in superspace, we define a new basis for the algebra of symmetric functions in superspace. On the other hand, we also show that sNSym can be realised as a Hopf algebra of trees.