Séminaire Lotharingien de Combinatoire, 93B.75 (2025), 12 pp.
Ayah Almousa, Shiliang Gao and Daoji Huang
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 characters of cyclic Demazure modules, solving a problem posted by Lam.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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