From Generalized Permutahedra to Grothendieck Polynomials via Flow Polytopes

Karola Mészáros ; Avery St. Dizier

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

We prove that for permutations 1π' where π' is dominant, the Grothendieck polynomial G_1π'(x) has saturated Newton polytope and that the Newton polytope of each homogeneous component of G_1π'(x) is a generalized permutahedron. We connect these Grothendieck polynomials to generalized permutahedra via a family of dissections of flow polytopes. We naturally label each simplex in a dissection by an integer sequence, called a left-degree sequence, and show that the sequences arising from simplices of a fixed dimension in our dissections of flow polytopes are exactly the integer points of generalized permutahedra. This connection of left-degree sequences and generalized permutahedra together with the connection of left-degree sequences and Grothendieck polynomials established in earlier work of Escobar and the first author reveals a beautiful relation between generalized permutahedra and Grothendieck polynomials.