Galois groups of Schubert problems via homotopy computation

Abstract

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in $\mathbb {C}^8$ meeting 15 fixed 5-planes non-trivially is the full symmetric group $S_{6006}$.