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Intersections of Schubert varieties and other permutation array schemes

  • Sara Billey
  • Ravi Vakil
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)

Abstract

Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. The number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach in the case of triple intersections). These equations may be used to compute Galois and monodromy groups of intersections of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of Eriksson and Linusson, and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a Littlewood-Richardson coefficient to be 0. We describe pathologies of Eriksson and Linusson’s permutation array varieties (failure of existence, irreducibility, equidimensionality, and reducedness of equations), and define the more natural permutation array schemes. In particular, we give several counterexamples to the Readability Conjecture based on classical projective geometry. Finally, we give examples where Galois/monodromy groups experimentally appear to be smaller than expected.

Key words

Schubert varieties permutation arrays Littlewood-Richardson coefficients 

AMS(MOS) subject classifications

Primary 14M17 Secondary 14M15 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Sara Billey
    • 1
  • Ravi Vakil
    • 2
  1. 1.Department of MathematicsUniversity of WashingtonSeattle
  2. 2.Department of MathematicsStanford UniversityStanford

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