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Mathematics of Computation

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Numerical Schubert calculus via the Littlewood-Richardson homotopy algorithm


Authors: Anton Leykin, Abraham Martín del Campo, Frank Sottile, Ravi Vakil and Jan Verschelde
Journal: Math. Comp. 90 (2021), 1407-1433
MSC (2020): Primary 14N15, 65H10
DOI: https://doi.org/10.1090/mcom/3579
Published electronically: February 4, 2021
MathSciNet review: 4232229
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Abstract: We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.


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Additional Information

Anton Leykin
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
MR Author ID: 687160
ORCID: 0000-0002-9216-3514
Email: leykin@math.gatech.edu

Abraham Martín del Campo
Affiliation: Centro de Investigación en Matemáticas, A.C., Jalisco S/N, Col. Valenciana, 36023, Guanajuato, Gto. México
ORCID: 0000-0003-0369-0652
Email: abraham.mc@cimat.mx

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
MR Author ID: 355336
ORCID: 0000-0003-0087-7120
Email: sottile@math.tamu.edu

Ravi Vakil
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
MR Author ID: 606760
ORCID: 0000-0001-8506-270X
Email: vakil@math.stanford.edu

Jan Verschelde
Affiliation: Deparment of Mathematics, Statisitics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, Illinois 60607
MR Author ID: 311807
Email: jan@math.uic.edu

Keywords: Schubert calculus, Grassmannian, Littlewood-Richardson rule, numerical homotopy continuation
Received by editor(s): April 17, 2018
Received by editor(s) in revised form: July 3, 2020
Published electronically: February 4, 2021
Additional Notes: This project was supported by American Institute of Mathematics through their SQuaREs program. The work of the first author was supported in part by the National Science Foundation under grant DMS-1719968. The work of the second author was supported in part by CONACyT under grant Cátedra-1076. The work of the third author was supported in part by the National Science Foundation under grant DMS-1501370. The work of the fourth author was supported in part by the National Science Foundation under grant DMS-1500334. The work of the fifth author was supported in part by the National Science Foundation under grants ACI-1440534 and DMS-1854513
Article copyright: © Copyright 2021 by the authors