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 Gr\"obner basis. Furthermore, we show that promotion and evacuation biject standard monomials of a positroid variety with those of its cyclic shifts and -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.
Keywords
Cite
@article{arxiv.2309.15384,
title = {Standard Monomials for Positroid Varieties},
author = {Ayah Almousa and Shiliang Gao and Daoji Huang},
journal= {arXiv preprint arXiv:2309.15384},
year = {2024}
}
Comments
48 pages; significantly rewritten. Section 6 now includes a connection to evacuation of tableaux. New Section 7 proves a formula for cyclic Demazure characters. New Appendix includes a result on Coxeter groups of potentially independent interest