Intersections of Schubert varieties and other permutation array schemes

Abstract

Using a blend of combinatorics and geometry, we give an algorithm foralgebraically finding all flags in any zero-dimensional intersection ofSchubert varieties with respect to three transverse flags, and more generally,any number of flags. In particular, the number of flags in a tripleintersection is also a structure constant for the cohomology ring of the flagmanifold. Our algorithm is based on solving a limited number of determinantalequations for each intersection (far fewer than the naive approach). Theseequations may be used to compute Galois and monodromy groups of intersectionsof Schubert varieties. We are able to limit the number of equations by usingthe permutation arrays of Eriksson and Linusson, and their permutation arrayvarieties, introduced as generalizations of Schubert varieties. We show thatthere exists a unique permutation array corresponding to each realizableSchubert problem and give a simple recurrence to compute the corresponding ranktable, giving in particular a simple criterion for a Littlewood-Richardsoncoefficient to be 0. We describe pathologies of Eriksson and Linusson'spermutation array varieties (failure of existence, irreducibility,equidimensionality, and reducedness of equations), and define the more naturalpermutation array schemes. In particular, we give several counterexamples tothe Realizability Conjecture based on classical projective geometry. Finally,we give examples where Galois/monodromy groups experimentally appear to besmaller than expected.