Curve neighborhoods of Schubert Varieties in the odd symplectic Grassmannian

TL;DR

This paper provides a comprehensive description of the irreducible components of curve neighborhoods of Schubert varieties in the odd symplectic Grassmannian, using combinatorial and algebraic tools, extending previous limited results.

Contribution

It introduces a full characterization of curve neighborhoods in the odd symplectic Grassmannian using Hecke products, partitions, and Bruhat order, advancing the understanding of these geometric structures.

Findings

Complete description of irreducible components of curve neighborhoods

Connection between Hecke products and Schubert varieties

Extension of previous results to the odd symplectic case

Abstract

Let $\mbox{IG}(k,2n+1)$ be the odd symplectic Grassmannian. It is a quasi-ho\-mo\-ge\-neous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in $\mbox{IG}(k,2n+1)$ was used by Mihalcea and the second named author to prove an (equivariant) quantum Chevalley rule. In this paper we give a full description of the irreducible components of curve neighborhoods in terms of the Hecke product of (appropriate) Weyl group elements, $k$-strict partitions, and BC-partitions. The latter set of partitions respect the Bruhat order with inclusions. Our approach follows the philosophy of Buch and Mihalcea's curve neighborhood calculations of Schubert varieties in the homogeneous cases.