Zero-one Schubert polynomials

Abstract

We prove that if $\sigma \in S_m$ is a pattern of $w\in S_n$, then we can express the Schubert polynomial ${\mathfrak{S}}_w$ as a monomial times ${\mathfrak{S}}_{\sigma }$ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar's orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on ${\mathfrak{S}}_w$ being zero-one. In this case, the Schubert polynomial ${\mathfrak{S}}_w$ is equal to the integer point transform of a generalized permutahedron.

17 pages, 2 figures; graphics updated and various typos corrected in v2. More minor fixes in v3