2010/08/25 by Allen Knutson, Knutson, Allen · 1 citation
Mathematics · #05E99 #14M15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AG #math.CO #math.KT #msc:05E99 #msc:14M15
paper · pdf · doi:10.48550/arxiv.1008.4302
30 pages; color helpful but not essential
arxiv created 2010/08/25 · arxiv updated 2010/08/26
Vakil studied the intersection theory of Schubert varieties in the Grassmannian in a very direct way: he degenerated the intersection of a Schubert variety Xmu and opposite Schubert variety Xnu to a union Xlambda, with repetition. This degeneration proceeds in stages, and along the way he met a collection of more complicated subvarieties, which he identified as the closures of certain locally closed sets. We show that Vakil's varieties are positroid varieties_, which in particular shows they are normal, Cohen-Macaulay, have rational singularities, and are defined by the vanishing of Plücker coordinates [Knutson-Lam-Speyer]. We determine the equations of the Vakil variety associated to a partially filled ``puzzle'' (building on the appendix to [Vakil]), and extend Vakil's proof to give a geometric proof of the puzzle rule from [Knutson-Tao '03] for equivariant Schubert calculus. The recent paper [Anderson-Griffeth-Miller] establishes (abstractly; without a formula) three positivity results in equivariant K-theory of flag manifolds G/P. We demonstrate one of these concretely, giving a corresponding puzzle rule.