Schur and Power Sum Polytopes

Aguiar and Ardila defined a Hopf monoidal structure on the collection of generalized permutahedra of all dimensions; and from it, constructed a Hopf algebra on the same polytopes, which is isomorphic to the Hopf algebra of symmetric functions, Sym. This endows each element of Sym with a formal sum of permutahedra, so that we can think of symmetric functions as members of McMullen’s polytope algebra. In this talk, we give geometric models for the Schur and power sum symmetric functions, when regarded as elements of the aforesaid polytope algebra. This is accomplished through a combinatorial rule for the former ones and in the way of an explicit description for the latter ones. We also characterize when the resulting geometric objects correspond to polytopes with missing faces. (Joint work with Carolina Benedetti and Mario Sanchez)