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.
2000 Mathematics Subject Classification. Primary 65H10; Secondary 14N15, 68W30.
Key words and phrases. continuation, geometric Littlewood-Richard\-son rule, Grassmannian, homotopies, numerical Schubert calculus, path following, polynomial system, Schubert problems.