Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

TL;DR

This paper computes initial ideals of tangent cones at fixed points of Schubert varieties in orthogonal Grassmannians, revealing their structure as Stanley-Reisner face rings linked to non-intersecting lattice paths.

Contribution

It provides explicit descriptions of initial ideals as square-free monomial ideals and characterizes the associated simplicial complexes in orthogonal Grassmannian Schubert varieties.

Findings

Initial ideals are square-free monomial ideals.

The associated complexes encode non-intersecting lattice paths.

Provides combinatorial descriptions of tangent cone ideals.

Abstract

We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.