Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs

TL;DR

This paper characterizes smooth positroid varieties using pattern avoidance, provides formulas for counting smooth positroids, and introduces combinatorial methods for tangent space dimension and Bruhat interval characterization.

Contribution

It introduces new pattern avoidance criteria for smoothness, counts smooth positroids with formulas and q-analogs, and develops combinatorial tools for tangent space and Bruhat interval analysis.

Findings

Characterization of smooth positroid varieties via pattern avoidance.

Formulas and q-analogs for counting smooth positroids.

A combinatorial method for tangent space dimension and Bruhat interval description.

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. This allows us to give two formulas for counting the number of smooth positroids along with two $q$-analogs. 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.