Gröbner Bases for Positroid Varieties

One way to probe the structure of an algebraic variety is to understand the structure of its coordinate ring at each degree, which can be achieved by constructing a Gröbner basis. Influential work of Hodge in the 1940s paved the way for using Gröbner bases to combinatorially study the Grassmannian. In this talk, we will follow Hodge's approach in order to investigate certain subvarieties of the Grassmannian called positroid varieties. Introduced by Knutson--Lam--Speyer in 2013, positroid varieties provide a stratified decomposition of the Grassmannian into subvarieties which enjoys many advantages over other previously studied decompositions. We will see that a Gröbner basis approach to investigating positroid varieties has powerful applications in algebra, geometry, and combinatorics. This is joint work with Shiliang Gao (University of Illinois at Urbana - Champaign) and Daoji Huang (University of Minnesota - Twin Cities).