Pattern Avoidance Criteria for Smoothness of Positroid Varieties Via Decorated Permutations, Spirographs, and Johnson Graphs

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 thesis, 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. We also include results of Christian Krattenthaler, which give additional formulas for counting smooth positroids and the coeffcients of our q-analogs as well as an asymptotic growth formula for the number of smooth 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. The results and much of the text of this thesis appear in joint work with Sara Billey [BW22a; BW22b]. The enumerative results due to Krattenthaler presented here were inspired by conjectures and results announced in [BW22a]. His results are included as an appendix to [BW22b].