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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.
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Quotients of Uniform Positroids
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.
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- Award ID(s):
- 1802986
- PAR ID:
- 10326787
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 29
- Issue:
- 1
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.37236/10056
