Standard Monomials for Positroid Varieties

Abstract

We give an explicit characterization of the standard monomials for positroid varieties with respect to the Hodge degeneration and give a Gröbner 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.

The old Lemma 7.27 had a false statement. This has now been corrected by the new Lemma 7.29 and proved by Appendix B on distributive properties of ideals, which was added. To appear in Advances in Mathematics