An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

TL;DR

This paper introduces a polynomial-time algorithm to determine if a Schubert polynomial coefficient vanishes, using a tableau criterion and new geometric characterizations, contrasting with the computational hardness of explicit coefficient calculation.

Contribution

It provides the first polynomial-time decision algorithm for the vanishing of Schubert polynomial coefficients based on a novel tableau criterion and Schubitope characterization.

Findings

The tableau criterion effectively decides coefficient vanishing.

Schubitope generalizes the permutahedron for subset grids.

Explicit coefficient computation remains #P-complete.

Abstract

Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid. In contrast, we show that computing these coefficients explicitly is #P-complete.