Newton polytopes in algebraic combinatorics

Abstract

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield-Pólya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others.Our principal construction is the Schubitope. For any subset of [n] 2 , we describe it by linear inequalities. This generalized permutahedron conjecturally has positive Ehrhart polynomial. We conjecture it describes the Newton polytope of Schubert and key polynomials. We also define dominance order on permutations and study its poset-theoretic properties.

Schubert polynomials as integer point transforms of generalized permutahedra

“…This confirms Conjectures 3.10 and 5.1 of [16], namely that key polynomials and Schubert polynomials have SNP. We now confirm Conjectures 3.9 and 5.13 of [16], which give a conjectural inequality description for the Newton polytopes of Schubert and key polynomials. We state this description and match it to the Minkowski sum description proven in Theorem 7.…”

“…After reviewing the necessary background, we prove our main theorem and draw corollaries about Schubert and key polynomials, confirming several recent conjectures of Monical, Tokcan and Yong [16].…”

“…The Newton polytope Newton(f ) ⊆ R I is the convex hull of the support of f . Following the definition of [16], we say that a polynomial f has saturated Newton polytope (SNP) if every lattice point in Newton(f ) is a vector in the support of f . In other words, SNP means that the polynomial is equal to a positively weighted integer point transform of its Newton polytope.…”

From generalized permutahedra to Grothendieck polynomials via flow polytopes

“…In Theorem B, we show that the Newton polytope of every homogeneous component of a left-degree polynomial is a generalized permutahedron. We also prove the saturated Newton polytope property (SNP) of Monical, Tokcan, and Yong [14]: every integer point in the Newton polytope is in the support of the polynomial.…”

Newton polytopes and symmetric Grothendieck polynomials

“…, x n ] is the convex hull of its exponent vectors, i.e., Newton(f ) = conv({α : c α = 0}) ⊆ R n . In [MoToYo17], f is said to have saturated Newton polytope (SNP) if c α = 0 whenever α ∈ Newton(f ). A study of SNP and algebraic combinatorics was given in loc.…”