From generalized permutahedra to Grothendieck polynomials via flow polytopes

Abstract

We study a family of dissections of flow polytopes arising from the subdivision algebra. To each dissection of a flow polytope, we associate a polynomial, called the left-degree polynomial, which we show is invariant of the dissection considered (proven independently by Grinberg). We prove that left-degree polynomials encode integer points of generalized permutahedra. Using that certain left-degree polynomials are related to Grothendieck polynomials, we resolve special cases of conjectures by Monical, Tokcan, and Yong regarding the saturated Newton polytope property of Grothendieck polynomials.

Lorentzian polynomials from polytope projections

“…We note that a special case of Proposition 3.4 was considered in [7,Section 4]. G) corresponding to an edge connecting i to n + 1.…”

“…By definition, the support of a Lorentzian polynomial forms the integer points of a generalized permutahedron. Brändén and Huh show that the normalization of the integer point transform of a generalized permutahedron is always Lorentzian [3, Theorem 7.1(4), (7)]. When these generalized permutahedra have vertices in {0, 1} n , certain projections of their integer point transforms are also Lorentzian, by [3,Theorem 2.10] and [3,Corollary 6.7].…”

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

“…Flow polytopes have been used in various fields like toric geometry [14] and representation theory [2]. More recently, they have been related to geometric and algebraic combinatorics thanks to connections with Schubert polynomials [10], diagonal harmonics [21], Gelfand-Tsetlin polytopes [18], and generalized permutahedra [22].…”

“…In [22] the authors give an explicit bijection between the maximal cliques and the integer flows. The map from the cliques to the flows is as follows.…”

“…The inverse map Ω −1 G, can be found in [22,Section 7]) (denoted Λ G, ). A framing (G, ) induces the following framing on the reverse graph G r : for an internal vertex i, let r in(i) := out(i) and r out(i) := in(i) .…”