Standard Monomials for Positroid Varieties

TL;DR

This paper characterizes standard monomials for positroid varieties, connects them with crystal structures, and provides formulas for their Hilbert series and characters, advancing algebraic and combinatorial understanding of these geometric objects.

Contribution

It offers an explicit description of standard monomials for positroid varieties, links them to crystal bases, and derives inductive formulas for their characters.

Findings

Explicit characterization of standard monomials for positroid varieties

Identification of monomials with Lam's cyclic Demazure crystal

Inductive formula for the character of cyclic Demazure modules

Abstract

We give an explicit characterization of the standard monomials for positroid varieties with respect to the Hodge degeneration and give a Gr\"obner basis. Furthermore, we show that promotion and evacuation biject standard monomials of a positroid variety with those of its cyclic shifts and $w_0$-reflection, respectively. The connection to promotion allows us to identify standard monomials of a positroid variety with Lam's cyclic Demazure crystal. Using a recurrence on the Hilbert series, we give an inductive formula for the character of cyclic Demazure modules.