Séminaire Lotharingien de Combinatoire, 80B.89 (2018), 12 pp.
Karola Mészáros and Avery St. Dizier
From Generalized Permutahedra to Grothendieck Polynomials via Flow Polytopes
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
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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