[Paper Review] Newton polytopes and symmetric Grothendieck polynomials

Laura Escobar, Alexander Yong|arXiv (Cornell University)|May 22, 2017

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper proves that symmetric Grothendieck polynomials and their homogeneous components have saturated Newton polytopes (SNP), with each homogeneous component's Newton polytope being a permutahedron. The key result establishes that the Newton polytope of each degree-$k$ component is the permutahedron of a specific partition $μ^{(k)}$, resolving conjectures by Monical-Tokcan-Yong and Fink-Mészáros-St. Dizier in the symmetric case.

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.

Motivation & Objective

• To resolve Conjecture 5.5 of Monical-Tokcan-Yong regarding the Newton polytope of symmetric Grothendieck polynomials for Grassmannian permutations.
• To verify Conjecture 5.1 of Fink-Mészáros-St. Dizier on the Newton polytope of symmetric Grothendieck polynomials in the case of dominant permutations.
• To establish that the Newton polytope of each homogeneous component of a symmetric Grothendieck polynomial is a permutahedron.
• To prove that symmetric Grothendieck polynomials have saturated Newton polytopes (SNP), meaning all lattice points in the Newton polytope correspond to non-zero monomials.

Proposed method

• Define a sequence of partitions $\mu^{(k)}$ by iteratively adding boxes to the highest possible rows under the constraint $\mu^{(k)}_r - \lambda_r < r-1$.
• Show that $\mu^{(k)}$ is the dominance-maximal shape for which the structure coefficient $a_{\lambda,\mu}$ is non-zero in the expansion of $G_\lambda$ in Schur polynomials.
• Apply Rado's theorem to equate dominance order with inclusion of permutahedra: $\mathcal{P}_\theta \subseteq \mathcal{P}_\delta$ iff $\theta \leq_D \delta$.
• Use convex combinations of exponent vectors and majorization to show that any lattice point in the Newton polytope of $G_\lambda$ lies within the permutahedron $\mathcal{P}_{\mu^{(k)}}$ for the corresponding degree $k$.
• Prove that the Newton polytope of $G_\lambda$ is the union of the permutahedra $\mathcal{P}_{\mu^{(k)}}$ for $k = 0$ to $N$, and that all lattice points in this union correspond to non-zero monomials.
• Leverage the fact that $\overline{\mu} = \sum c_k \mu^{(k)}$ is majorized by $\mu^{(K)}$, ensuring that any convex combination of exponent vectors lies within the Newton polytope of the degree-$K$ component.

Experimental results

Research questions

• RQ1 Do symmetric Grothendieck polynomials have saturated Newton polytopes (SNP), meaning all lattice points in the Newton polytope correspond to non-zero monomials?
• RQ2 Is the Newton polytope of each homogeneous component of a symmetric Grothendieck polynomial a permutahedron?
• RQ3 Does the dominance order of partitions correspond to inclusion of their associated permutahedra in the context of Grothendieck polynomials?
• RQ4 Can the structure coefficients $a_{\lambda,\mu}$ be characterized combinatorially via the maximality of $\mu^{(k)}$ under dominance order?
• RQ5 Do the conjectures of Monical-Tokcan-Yong and Fink-Mészáros-St. Dizier hold for symmetric Grothendieck polynomials in the Grassmannian and dominant cases?

Key findings

• The Newton polytope of the symmetric Grothendieck polynomial $G_\lambda(x_1,\ldots,x_n)$ is saturated, meaning every lattice point in the convex hull of its exponent vectors corresponds to a non-zero monomial.
• Each homogeneous component $G_\lambda[k]$ of degree $k$ has a Newton polytope equal to the permutahedron $\mathcal{P}_{\mu^{(k)}}$, where $\mu^{(k)}$ is the partition obtained by successively adding boxes under the given constraints.
• The construction of $\mu^{(k)}$ yields the dominance-maximal shape for which the coefficient $a_{\lambda,\mu}$ is non-zero in the Schur expansion of $G_\lambda$.
• The Newton polytope of $G_\lambda$ is the union of the permutahedra $\mathcal{P}_{\mu^{(k)}}$ for $k = 0$ to $N$, and this union is saturated.
• The proof relies on Rado's theorem to relate dominance order to permutahedron inclusion, and majorization arguments to show that convex combinations of exponent vectors remain within the Newton polytope.
• The result confirms Conjecture 5.5 of Monical-Tokcan-Yong and Conjecture 5.1 of Fink-Mészáros-St. Dizier in the symmetric case, extending prior results to the Grassmannian and dominant permutation settings.