Galois groups of Schubert problems via homotopy computation
This work provides a computational approach to a pure mathematics problem, demonstrating that Galois groups in Schubert calculus can be determined via homotopy continuation, which is a novel application for this domain.
The authors used numerical homotopy continuation to compute Galois groups in the Schubert calculus, showing that the Galois group for a specific Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.
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 C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.