Newton polytopes in algebraic combinatorics

Cara Monical, Neriman Tokcan, Alexander Yong

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Article number66
JournalSelecta Mathematica, New Series
Volume25
Issue number5
DOIs
StatePublished - Dec 1 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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