Newton Polytopes in Algebraic Combinatorics

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