Standard Monomials for Positroid Varieties

Influential work of Hodge from the 1940s led the way in using Gröbner bases to combinatorially study the Grassmannian. We follow Hodge's approach to investigate certain subvarieties of the Grassmannian, 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 present an explicit combinatorial description of the standard monomials and Gröbner bases of positroid varieties under the Hodge degeneration. As applications, we establish a connection between cyclic and reflectional symmetry of positroid varieties and well-known combinatorial operations on semistandard Young tableaux. We also give a formula for computing the characters of cyclic Demazure modules by a recurrence on the Hilbert series of positroid varieties, answering a question of Lam.

This is based on joint work with Ayah Almousa and Shiliang Gao.