Ayah Almousa
University of Kentucky
Title
Gorenstein positroid varieties
Abstract
Positroid varieties are an important class of subvarieties of the Grassmannian which carry rich combinatorial structures from both Schubert geometry and matroid theory. Knutson--Lam--Speyer showed that they are normal and Cohen--Macaulay with rational singularities, and Billey--Weaver gave criteria for smoothness. In this talk, I will describe a combinatorial characterization of Gorenstein positroid varieties. The key ingredient is a description of their Picard groups: when the underlying matroid is connected, every line bundle is a power of the Plücker line bundle. This lets us translate the Gorenstein condition into a system of integer linear equations and then into a condition on cycles in a graph associated to a bounded affine permutation. I will explain the geometric and combinatorial ideas behind this criterion and illustrate it with examples. This is joint work with Anders Buch and Daoji Huang.
