[Paper Review] Positroid varieties I: juggling and geometry

Allen Knutson, Thomas Lam+1 more|ArXiv.org|Mar 21, 2009

Advanced Combinatorial Mathematics28 references62 citations

TL;DR

This paper introduces positroid varieties as intersections of cyclically shifted Bruhat cells in the Grassmannian, establishing they are normal, Cohen-Macaulay, and defined by vanishing Pl"ucker coordinates. The key contribution is a new indexing via bounded juggling patterns (affine Weyl group elements), linking them to affine Stanley functions and quantum Schubert calculus.

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.

Motivation & Objective

• To define and study a new stratification of the Grassmannian that refines the Richardson stratification while preserving desirable geometric properties.
• To resolve the intractability of the full GGMS decomposition by restricting to cyclic shifts of a single Bruhat decomposition.
• To provide a cyclic-invariant, combinatorially tractable indexing of strata using bounded juggling patterns in the affine Weyl group.
• To establish that positroid varieties are normal, Cohen-Macaulay, and defined by Pl"ucker relations, with computable equivariant Hilbert series.
• To connect Postnikov's positroid approach with Buch-Kresch-Tamvakis' quantum Schubert calculus via affine Stanley functions.

Proposed method

• Define positroid varieties as the intersection of $n$ cyclically shifted Schubert varieties, showing they coincide with closures of intersections of cyclically shifted Bruhat cells.
• Introduce bounded juggling patterns as a new combinatorial indexing set for positroid varieties, derived from affine permutations.
• Apply Frobenius splitting techniques from Brion, Lakshmibai, and Littelmann to prove normality and rational singularities.
• Use Hodge-Gr"obner degenerations to show each positroid variety degenerates to a projective Stanley-Reisner scheme of a shellable ball.
• Compute the $T$-equivariant Hilbert series of positroid varieties using the theory of affine Stanley functions.
• Establish a geometric link between positroid varieties and quantum Schubert calculus by showing their cohomology classes are represented by affine Stanley functions.

Experimental results

Research questions

• RQ1 Can a refinement of the Richardson stratification be constructed that preserves normality, Cohen-Macaulayness, and rational singularities, while avoiding the pathologies of the GGMS decomposition?
• RQ2 What combinatorial structure provides a cyclic-invariant indexing of the strata in such a refined decomposition?
• RQ3 How are positroid varieties related to quantum Schubert calculus and affine Stanley functions?
• RQ4 Can the cohomology classes of positroid varieties be explicitly described using symmetric function theory?
• RQ5 What is the geometric behavior of positroid varieties under degeneration, and how does this relate to their singularities?

Key findings

• Positroid varieties are normal and Cohen-Macaulay with rational singularities, as established via Frobenius splitting of associated Richardson varieties.
• Each positroid variety is defined as a scheme by the vanishing of certain Pl"ucker coordinates, confirming their scheme-theoretic structure.
• The $T$-equivariant Hilbert series of a positroid variety is given by the corresponding affine Stanley function, linking geometry to symmetric function theory.
• The cohomology class of a positroid variety is represented by an affine Stanley function, enabling a direct connection to Postnikov's and Buch-Kresch-Tamvakis' approaches in quantum Schubert calculus.
• Each positroid variety degenerates under a Hodge-Gr"obner degeneration to the projective Stanley-Reisner scheme of a shellable ball, confirming their topological regularity.
• The map from the flag manifold to the Grassmannian projects Richardson varieties onto positroid varieties, and this projection preserves the strata structure, as shown via the theorem on pullbacks of Schubert varieties.