# Classification of Schubert Galois groups in Gr(4,9)

## TL;DR

This paper classifies the Galois groups of Schubert problems in Gr(4,9), revealing that only a small subset have non-full symmetric groups, which are imprimitive and fall into two geometric families.

## Contribution

It provides a detailed classification of the Galois groups for all essential Schubert problems in Gr(4,9), identifying the 149 with non-full symmetric groups and analyzing their structure.

## Key findings

- 149 Schubert problems have Galois groups not containing the alternating group.
- All such Galois groups are imprimitive permutation groups.
- The 149 problems fall into two geometric families.

## Abstract

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149 -- each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and it begins to establish the inverse Galois problem for Schubert calculus.