#####
Séminaire Lotharingien de Combinatoire, 85B.90 (2021), 12 pp.

# Eric Marberg and Brendan Pawlowski

# Gröbner Geometry for Skew-Symmetric Matrix Schubert Varieties

**Abstract.**
Matrix Schubert varieties are the orbit closures of *B* × *B* acting on all *n* × *n* matrices, where *B* is the group of invertible lower triangular matrices. We define skew-symmetric matrix Schubert varieties to be the orbit closures of *B* acting on all *n* × *n* skew-symmetric matrices.
In analogy with Knutson and Miller's work on matrix Schubert varieties, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams, analogous to the pipe dreams of Bergeron and Billey. We show that these initial ideals are the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

The following versions are available: