The extra basis in noncommuting variables

TL;DR

This paper provides a combinatorial description of the coproduct in the noncommutative symmetric functions algebra, introduces a new basis, and connects Hopf monoids with Grothendieck bialgebras.

Contribution

It offers the first combinatorial description of the coproduct of the x-basis in NCSym and introduces a new multiplicative basis for symmetric functions.

Findings

Combinatorial description of the coproduct of the x-basis in NCSym.

New multiplicative basis for the algebra of symmetric functions.

Connections established between Hopf monoids, Fock functor, and NCSym.

Abstract

We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the x-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this using the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the x-basis we discover a new multiplicative basis for the algebra of symmetric functions.