Solving schubert problems with Littlewood-Richardson homotopies

Authors:
Frank Sottile;Ravi Vakil;Jan Verschelde
Affiliations:
Texas A&M University, College Station, TX;Stanford University, Stanford, CA;University of Illinois at Chicago, Chicago, IL
Venue:
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Year:
2010

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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.