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.
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Acknowledgements
We thank Alexander Barvinok, Laura Escobar, Sergey Fomin, Allen Knutson, Melinda Lanius, Fu Liu, Mark Shimozono, John Stembridge, Sue Tolman and Anna Weigandt for very helpful conversations. We thank Bruce Reznick specifically for his example of \(f=x_1^2+x_2 x_3+\cdots \) we used in the introduction. AY was supported by an NSF grant. CM and NT were supported by UIUC Campus Research Board Grants. We made significant use of SAGE during our investigations.
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