skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: A Pattern Avoidance Characterization for Smoothness of Positroid Varieties
Positroids are certain representable matroids originally studied by Post- nikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defin- ing positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson– Lam–Speyer and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids. Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at key points using an induced subgraph of the Johnson graph. We also give a Bruhat interval char- acterization of positroids.  more » « less
Award ID(s):
1764012
PAR ID:
10377699
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Séminaire lotharingien de combinatoire
Volume:
86
Issue:
B
ISSN:
1286-4889
Page Range / eLocation ID:
Article #57
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; ; ; (, Pacific Journal of Mathematics)
    Positroid strata of the complex Grassmannian can be realized as augmentation varieties of Legendrians called positroid links. We prove that the partial order on strata induced by Zariski closure also has a symplectic interpretation, given by exact Lagrangian cobordism. 
    more » « less
  2. ; (, International Mathematics Research Notices)
    null (Ed.)
    Abstract Tropical geometry and the theory of Newton–Okounkov bodies are two methods that produce toric degenerations of an irreducible complex projective variety. Kaveh and Manon showed that the two are related. We give geometric maps between the Newton–Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of $$X$$. Under a technical condition, we produce a natural “algebraic wall-crossing” map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian $Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one $$T$$-varieties; Ilten also explains the connection to the “combinatorial mutations” studied by Akhtar–Coates–Galkin–Kasprzyk. 
    more » « less
  3. ; ; (, The Electronic Journal of Combinatorics)
    Two matroids $$M$$ and $$N$$ are said to be concordant if there is a strong map from $$N$$ to $$M$$. This also can be stated by saying that each circuit of $$N$$ is a union of circuits of $$M$$. In this paper, we consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorics to determine concordance among some of them. More precisely, given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that is concordant with it. We do this by means of their associated decorated permutations. As a byproduct of our work, we describe completely the collection of circuits of this particular subset of positroids. 
    more » « less
  4. (, International Mathematics Research Notices)
    Abstract We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_f$$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of $$\operatorname{Crit}^{\geqslant 0}_f$$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_{k,n}$$ is homeomorphic to the second hypersimplex $$\Delta _{2,n}$$. 
    more » « less
  5. (, Canadian Mathematical Bulletin)
    Abstract We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email