Solving Schubert Problems with Littlewood-Richardson Homotopies

TL;DR

This paper introduces a novel numerical homotopy continuation method based on Littlewood-Richardson rules for efficiently solving all solutions to Schubert problems on Grassmannians, leveraging geometric insights and optimized path tracking.

Contribution

It develops the Littlewood-Richardson homotopy algorithm, a new approach that uses geometric proofs and checker configurations to solve Schubert problems numerically.

Findings

Efficiently finds all solutions to Schubert problems.

Optimizes the number of paths tracked for generic problems.

Implemented in PHCpack for practical use.

Abstract

We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.