Galois groups of Schubert problems of lines are at least alternating

TL;DR

This paper proves that the Galois groups of Schubert problems involving lines in projective space are at least the alternating group, expanding understanding of the symmetry properties of these enumerative geometric problems.

Contribution

It establishes that the Galois group of any such Schubert problem contains the alternating group, using combinatorial and integral techniques to prove key inequalities.

Findings

Galois groups contain the alternating group for all such problems

Largest family of enumerative problems with partially determined Galois groups

Uses combinatorial injections and integral formulas to establish inequalities

Abstract

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formula to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.