Positroid varieties I: juggling and geometry
Abstract
While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the {\em cyclic shifts} of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, and BrownGoodearlYakimov. However, its cyclicinvariance is hidden in this description. Postnikov gave many cyclicinvariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call {\em bounded juggling patterns}. We adopt his terminology and call the strata {\em positroid varieties.} We show that positroid varieties are normal and CohenMacaulay, and are defined as schemes by the vanishing of Plucker coordinates. We compute their Tequivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and BuchKreschTamvakis' approaches to quantum Schubert calculus. Our principal tools are the Frobenius splitting results for Richardson varieties as developed by Brion, Lakshmibai, and Littelmann, and the HodgeGrobner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective StanleyReisner scheme of a shellable ball.
 Publication:

arXiv eprints
 Pub Date:
 March 2009
 arXiv:
 arXiv:0903.3694
 Bibcode:
 2009arXiv0903.3694K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics
 EPrint:
 58 pages