Séminaire Lotharingien de Combinatoire, 80B.89 (2018), 12 pp.

Karola Mészáros and Avery St. Dizier

From Generalized Permutahedra to Grothendieck Polynomials via Flow Polytopes

Abstract. We prove that for permutations 1π' where π' is dominant, the Grothendieck polynomial G1π'(x) has saturated Newton polytope and that the Newton polytope of each homogeneous component of G1π'(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.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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