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Positroid varieties: juggling and geometry

Part of: Designs and configurations Special varieties Algebraic combinatorics

Published online by Cambridge University Press:  19 August 2013

Allen Knutson ,
Thomas Lam  and
David E. Speyer
Allen Knutson
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA email [email protected]
Thomas Lam
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email [email protected]@umich.edu
David E. Speyer
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email [email protected]@umich.edu
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Abstract

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has 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, Brown–Goodearl–Yakimov and the present authors. 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 bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove 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.

Keywords

Grassmannianpositroidaffine Stanley polynomialquantum cohomology

MSC classification

Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds 05B35: Matroids, geometric lattices 05E05: Symmetric functions and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 10 , October 2013 , pp. 1710 - 1752
Copyright
© The Author(s) 2013 

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