Influential work of Hodge from the 1940s led the way in using Gröbner bases to combinatorially study the Grassmannian. In this talk, we will follow Hodge's approach in order to investigate certain subvarieties called positroid varieties. Positroid varieties, introduced by Knutson-Lam-Speyer in 2013, provide a stratification of the Grassmannian which enjoys many advantages over other previously studied decompositions. In this talk, we will present an explicit and combinatorial description of the standard monomials and Gröbner bases of positroid varieties under the Plücker embedding.  As an application, we establish a connection between promotion on semi-standard Young tableaux and cyclic symmetry of positroid varieties.

This is joint work with Ayah Almousa and Shiliang Gao.