Séminaire Lotharingien de Combinatoire, 86B.57 (2022), 12 pp.
Sara Billey and Jordan E. Weaver
A Pattern Avoidance
Characterization for Smoothness of Positroid Varieties
Abstract.
Positroids are certain representable matroids originally
studied by Postnikov in connection with the totally nonnegative
Grassmannian and now used widely in algebraic combinatorics. The
positroids give rise to determinantal equations defining 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 characterization
of positroids.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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