Galois groups of Schubert problems via homotopy computation

TL;DR

This paper applies numerical homotopy continuation methods to compute Galois groups in Schubert calculus, demonstrating the approach's effectiveness through explicit examples like the Galois group of a specific Schubert problem.

Contribution

It introduces a novel computational approach using numerical homotopy continuation to determine Galois groups in pure mathematics, bridging numerical methods and algebraic geometry.

Findings

Galois group of a specific Schubert problem is the full symmetric group S_6006.

Numerical homotopy continuation effectively computes Galois groups in algebraic geometry.

The method provides concrete computational evidence for Galois group structures.

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 C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.