Positroid varieties: juggling and geometry

By Allen Knutson, Thomas Lam, David E. Speyer. Compositio Mathematica, 2009

235 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f731fc5d0b38f1c9a37366fb42a79c51a9b03663

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Citations in the corpus

The twist for positroid varieties (2016, background): “Knutson, Lam and Speyer identified the corresponding complex varieties in work that appeared on the arXiv in 2009 [14] and in improved form in 2011 [15] (publication date 2013).”
Back stable Schubert calculus (2018, background): “ Grassmannians where it can be identiﬁed (after equivariant shifts) with graph Schubert varieties, a special class of positroid varieties. This connects our work with that of Knutson, Lam, and Speyer [KLS], who identiﬁed the equivariant cohomology classes of positroid varieties with aﬃne double Stan”
Cluster Algebras of Grassmannians are Locally Acyclic (2014, background): “oid cell ˚Π( w) to be the set of L in G(k,n) with r ij(L) = (j −i+1)−#([i,j] ∩ w([i,j])). Each ˚Π( w) is an irreducible, locally closed, algebraic variety. The dimension of ˚Π( w) is k(n−k)−ℓ(w). See [KLS13] for numerous alternate deﬁnitions of positroid cells. It is anticipated that the homogeneous”
Cluster varieties from Legendrian knots (2015, background): “eometry, for example double Bruhat cells and spaces of congurations of ags [KLS, BFZ,FG2]. We will show that one of the combinatorial formalisms for labelling strata { so-called cyclic rank matrices [KLS] { directly translates to a modular description of these strata in terms of Legendrian knots: Th”
Positive geometries and canonical forms (2017, background): “The canonical form Ω(f) := Ω(Πf ,Πf,≥0) was studied in [3, 11].”
Weak separation and plabic graphs (2011, background): “positroids While our results are purely combinatorial, they are motivated by constructions in algebraic geometry. Speciﬁcally, [9] introduced the positroid stratiﬁcation of the Grassmannian (see also [7]). We will prove a version of Theorem 3.3 for every cell in this stratiﬁcation. Theorem 3.3 itsel”
Cohomology of cluster varieties, I: Locally acyclic case (2016, background): “ASLAMANDDAVIDSPEYER 6.6. Grassmannians. The open positroid variety ˚Π( r,d) in the Grassmannian Gr(r,d) of r-planes in Cd is obtained by removing the nhyperplanes {∆ i,i+1,...,i+r−1 = 0} from Gr(r,d) [KLS], where ∆I denotes a Plu¨cker coordinate. The open positroid variety is a cluster variety [Sco]”
A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties (2016, methodology): “Our Chevalley formula is proved using an alternative version of the quantum equals classical theorem, which expresses the Gromov-Witten invariants of X in terms of projected Gromov-Witten varieties [39, 9].”
Regularity theorem for totally nonnegative flag varieties (2019, methodology): “which were introduced in [KLS13], building on the work of Postnikov [Pos07].”
KP solitons and total positivity for the Grassmannian (2011, background): “4 is closely related to results on cyclic rank matrices from [13].”