Galois Groups of Composed Schubert Problems

TL;DR

This paper studies how composing Schubert problems on different Grassmannians affects their Galois groups, revealing that the Galois group of the composition often remains imprimitive and providing algebraic proofs and evidence for a broader conjecture.

Contribution

It introduces a new algebraic proof for the product formula of solutions in composed Schubert problems and explores the structure of their Galois groups, including a conjecture on their imprimitive nature.

Findings

The Galois group of a composed Schubert problem is often a subgroup of a wreath product.

The product formula for solutions is proven algebraically.

Evidence suggests all composed Schubert problems have imprimitive Galois groups.

Abstract

Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of the original problems. This generalizes a construction discovered while classifying Schubert problems on the Grassmannian of 4-planes in C^9 with imprimitive Galois groups. We give an algebraic proof of the product formula. In a number of cases, we show that the Galois group of the composed Schubert problem is a subgroup of a wreath product of the Galois groups of the original problems, and is therefore imprimitive. We also present evidence for a conjecture that all composed Schubert problems have imprimitive Galois groups.