Positroid varieties I: juggling and geometry - Mathematics > Algebraic Geometry

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Abstract: While the intersection of the Grassmannian Bruhat decompositions for allcoordinate flags is an intractable mess, the intersection of only the {\emcyclic shifts} of one Bruhat decomposition turns out to have many of the goodproperties of the Bruhat and Richardson decompositions. This decompositioncoincides with the projection of the Richardson stratification of the flagmanifold, studied by Lusztig, Rietsch, and Brown-Goodearl-Yakimov. However, itscyclic-invariance is hidden in this description. Postnikov gave manycyclic-invariant ways to index the strata, and we give a new one, by a subsetof the affine Weyl group we call {\em bounded juggling patterns}. We adopt histerminology and call the strata {\em positroid varieties.} We show thatpositroid varieties are normal and Cohen-Macaulay, and are defined as schemesby the vanishing of Plucker coordinates. We compute their T-equivariant Hilbertseries, and show that their associated cohomology classes are represented byaffine Stanley functions. This latter fact lets us connect Postnikov-s andBuch-Kresch-Tamvakis- approaches to quantum Schubert calculus. Our principaltools are the Frobenius splitting results for Richardson varieties as developedby Brion, Lakshmibai, and Littelmann, and the Hodge-Grobner degeneration of theGrassmannian. We show that each positroid variety degenerates to the projectiveStanley-Reisner scheme of a shellable ball.

Autor: Allen Knutson, Thomas Lam, David E Speyer

Fuente: https://arxiv.org/