Multi-rigidity of Schubert classes in partial flag varieties

TL;DR

This paper investigates the multi-rigidity property of Schubert classes in rational homogeneous spaces, establishing conditions under which these classes are uniquely represented by Schubert varieties, especially in partial flag varieties.

Contribution

It proves multi-rigidity of Schubert classes in rational homogeneous spaces and characterizes multi-rigid classes in partial flag varieties of types A, B, and D.

Findings

Multi-rigidity holds for Schubert classes in these spaces.

Characterization of multi-rigid classes in specific types of partial flag varieties.

Rigidity properties are derived from generalized Grassmannians and simple cases.

Abstract

In this paper, we study the multi-rigidity problem in rational homogeneous spaces. A Schubert class is called multi-rigid if every multiple of it can only be represented by a union of Schubert varieties. We prove the multi-rigidity of Schubert classes in rational homogeneous spaces. In particular, we characterize the multi-rigid Schubert classes in partial flag varieties of type A, B and D. Moreover, for a general rational homogeneous space $G/P$, we deduce the rigidity and multi-rigidity from the corresponding generalized Grassmannians (correspond to maximal parabolics). When $G$ is semi-simple, we also deduce the rigidity and multi-rigidity from the simple cases.