  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.
