Puzzles, positroid varieties, and equivariant K-theory of Grassmannians

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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.