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.

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