Positroid varieties I: juggling and geometry

Details Positroid varieties I: juggling and geometry Positroid varieties I: juggling and geometry

2009-03-21

arxiv.org/abs/0903.3694v1

Mathematics

Algebraic Geometry

58 pages

Scientific paper

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 Brown-Goodearl-Yakimov. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant 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 Cohen-Macaulay, and are defined as schemes by the vanishing of Plucker coordinates. We compute their T-equivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' 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 Hodge-Grobner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective Stanley-Reisner scheme of a shellable ball.

Affiliated with

Also associated with

No associations

Rating

Positroid varieties I: juggling and geometry does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Positroid varieties I: juggling and geometry, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Positroid varieties I: juggling and geometry will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-190697