Newton polytopes in algebraic combinatorics

By C. Monical, Neriman Tokcan, Alexander Yong. Selecta Mathematica, 2017

65 citations recorded by the source corpus.

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[n]^2$$\end{document}, 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/616cff1770d1bc801b16f0e193017816efc2bf8a

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In Algebra and Number Theory

1. Karola M'esz'aros
2. Alex Fink
3. Avery St. Dizier
4. Eric Katz

In Discrete Mathematics and Combinatorics

1. Karola M'esz'aros
2. Alex Fink
3. Avery St. Dizier
4. Eric Katz

In Geometry and Topology

1. Alex Fink

Citations in the corpus

• Schubert polynomials as integer point transforms of generalized permutahedra (2017, background): “efficients. 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]. Date: June 19, 2017. Fink is partially supported by an Engineering and Physical Sciences Resea”