# The Hopf Monoid of Orbit Polytopes

## TL;DR

This paper introduces a new Hopf monoid of orbit polytopes invariant under symmetric group actions, linking them to integer compositions and noncommutative symmetric functions, expanding the algebraic framework of combinatorial objects.

## Contribution

It defines the Hopf monoid of orbit polytopes, explores its structure via normal equivalence and Fock functor, and connects it to compositions and noncommutative symmetric functions.

## Key findings

- Orbit polytopes correspond to integer compositions under normal equivalence.
- Applying the Fock functor yields a Hopf algebra of compositions.
- The character group relates to noncommutative symmetric functions.

## Abstract

Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, posets, and matroids. We introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret the Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.