Newton polytopes and symmetric Grothendieck polynomials

Abstract

Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial K-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case. arXiv:1705.07876v1 [math.CO]

On the support of Grothendieck polynomials

“…We prove Conjectures 1.2 and 1.4 for Grassmannian permutations. We begin by reviewing the main result of [5]. Recall a permutation w is Grassmannian if w has exactly one descent.…”

Lattice Polytopes from Schur and Symmetric Grothendieck Polynomials

“…The inflated symmetric Grothendieck polynomial indexed by λ and h is Escobar and Yong [12] showed that the symmetric Grothendieck polynomial G λ (x) has SNP and described the components of the Newton polytope associated to the homogeneous components of G λ (x). We extend the work of Escobar and Yong to G h,λ (x) and show that G h,λ (x) also has SNP.…”

“…Proof. In [11], the proof that G 1,λ (x) = G λ (x) has SNP does not depend on the inflation parameter h other than in [11, Claim A], which describes the structure of Newt(G λ (x)) arising from the homogeneous components of G λ (x). Using the description of the homogeneous components of G h,λ (x) from Proposition 3.9, the rest of the proof in [11] applies to arbitrary h ∈ Z ≥1 and shows that G h,λ (x) has SNP.…”

“…We denote these polytopes by Newt(s λ ) and Newt(G h,λ ), respectively. The polytopes Newt(s λ ) and Newt(G 1,λ ) have previously been studied by Monical, Tokcan, and Yong [19] and Escobar and Yong [12], but many open questions remain. We are particularly interested in determining when Newt(s λ ) and Newt(G h,λ ) are reflexive and when they have the integer decomposition property (IDP), both of which we define in Section 2.…”

Logarithmic concavity of Schur and related polynomials

“…The M-convexity of the support of G k w was conjectured in [MS17, Conjecture 5.1] and proved in [EY17] when w is a Grassmannian permutation. Conjecture 20 implies Conjecture 15 because the degree pwq homogeneous component of G w is the Schubert polynomial S w .…”