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 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.

58 pages

References in corpus (5)

• Total positivity, Grassmannians, and networks
• A Monoid for the Universal K-Bruhat Order
• Notes on Schubert classes of a loop group
• Finiteness of the number of compatibly-split subvarieties
• Positroids and Schubert matroids

Cited by in corpus (33)

• Weak Separation and Plabic Graphs
• Evidence for a Nonplanar Amplituhedron
• ABJM amplitudes and the positive orthogonal grassmannian
• Rationalizing Loop Integration
• The twist for positroid varieties
• $k$-Schur functions and affine Schubert calculus
• Projected Richardson varieties and affine Schubert varieties
• Puzzles, positroid varieties, and equivariant K-theory of Grassmannians
• Stanley symmetric functions and Peterson algebras
• On-Shell Structures of MHV Amplitudes Beyond the Planar Limit
• Multivariate Juggling Probabilities
• Period Integrals and the Riemann-Hilbert Correspondence
• The Deodhar decomposition of the Grassmannian and the regularity of KP solitons
• Cyclicity of Lusztig's stratification of Grassmannians and Poisson geometry
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms
• Affine Patches on Positroid Varieties and Affine Pipe Dreams (Thesis)
• Frobenius splitting and Möbius inversion
• Standard Monomial Theory for desingularized Richardson varieties in the flag variety GL(n)/B
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• The Positive orthogonal Grassmannian and loop amplitudes of ABJM
• Fomin-Greene monoids and Pieri operations
• Toric Poisson Ideals in Cluster Algebras
• A combinatorial proof that Schubert vs. Schur coefficients are nonnegative
• The Purity Conjecture in Type $C$
• On the totally positive grassmannian
• Bridge graphs and Deodhar parametrizations for positroid varieties
• Combinatorics of symmetric plabic graphs
• Infinite flags and Schubert polynomials
• Orientations of BCFW Charts on the Grassmannian
• Twisting the quantum grassmannian
• Geometric Shifts and Positroid Varieties
• Patch ideals and Peterson varieties