The Hopf monoid of orbit polytopes

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.

Keywords

Hopf monoids, generalized permutahedra, orbit polytopes, weight polytopes, noncommutative symmetric functions, integer compositions

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The author was supported by the Chancellor’s Fellowship of the University of California, Berkeley and the National Physical Science Consortium graduate fellowship.

Received 11 October 2019

Accepted 13 October 2019

Published 9 October 2020