Overview
This research investigates the Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties. The study specifically aimed to compute this regularity by deriving a combinatorial formula. Furthermore, it characterized the highest-degree homogeneous component associated with these varieties and determined their maximal Castelnuovo-Mumford regularity.
Research Context
Skew-symmetric matrix Schubert varieties are defined as determinantal varieties. They are formed by the intersection of matrix Schubert varieties with the space of skew-symmetric matrices. These varieties exhibit a close relationship with the orbit closures generated by the action of the symplectic group on the flag variety. The torus-equivariant K-classes associated with skew-symmetric matrix Schubert varieties are identified as symplectic Grothendieck polynomials.
Approach
The primary approach involved computing the Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties. This computation was achieved by developing a combinatorial formula for the degree of symplectic Grothendieck polynomials. In addition to this, the research characterized the highest-degree homogeneous component present within a symplectic Grothendieck polynomial. The study also proceeded to compute the maximal Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties.
Findings
- A combinatorial formula was developed for determining the degree of symplectic Grothendieck polynomials.
- This combinatorial formula was utilized to compute the Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties.
- The highest-degree homogeneous component of a symplectic Grothendieck polynomial was characterized.
- The maximal Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties was computed.