Restrictions on Galois groups of Schubert problems

TL;DR

This paper investigates the possible Galois groups of Schubert problems, expanding known cases on Gr(4,9) and identifying conditions that lead to smaller Galois groups than the full symmetric group.

Contribution

It fully explores Schubert problems on Gr(4,9) to determine their Galois groups and finds conditions that restrict the Galois group size in large Grassmannians.

Findings

Galois groups for Schubert problems on Gr(4,9) are now fully characterized.

Certain Schubert conditions imply the Galois group is smaller than the full symmetric group.

The study broadens understanding of Galois groups in algebraic geometry.

Abstract

The Galois group of a Schubert problem encodes some structure of its set of solutions. Galois groups are known for a few infinite families and some special problems, but what permutation groups may appear as a Galois group of a Schubert problem is still unknown. We expand the list of Schubert problems with known Galois groups by fully exploring the Schubert problems on $\text{Gr}(4,9)$, the smallest Grassmannian for which they are not currently known. We also discover sets of Schubert conditions for any sufficiently large Grassmannian that imply the Galois group of a Schubert problem is much smaller than the full symmetric group.