Restrictions on Galois Groups of Schubert Problems

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 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.

These results are attained by combining computational exploration with geometric arguments. We use a technique initially described by Vakil to filter out many problems whose Galois group contains the alternating group. We then implement a more computationally intensive algorithm that collects data about the Galois groups of the remaining problems. For each of these, we either gather enough data about elements in the Galois group to determine that it must be the full symmetric group, or we find structure in the set of solutions that restricts the Galois group. Combining the restrictions imposed by the structure of the solutions with the data gathered about the group through the algorithm, we are able to determine the Galois group of these problems as well.