An inequality of Kostka numbers and Galois groups of Schubert problems

TL;DR

This paper proves that the Galois group of any Schubert problem with lines in projective space contains the alternating group, by establishing a key inequality among Kostka numbers using combinatorial and spectral analysis methods.

Contribution

It introduces a novel inequality among Kostka numbers and applies spectral analysis to prove Galois group properties of Schubert problems involving lines.

Findings

Galois groups contain the alternating group

Established a new inequality among Kostka numbers

Applied spectral analysis to combinatorial problems

Abstract

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of sl_2(C)-modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.