Schur flexibility of cominuscule Schubert varieties

TL;DR

This paper characterizes Schur rigidity of Schubert classes in cominuscule rational homogeneous varieties, establishing a necessary and sufficient condition for Schur rigidity.

Contribution

It proves that a previously known sufficient condition for Schur rigidity is also necessary, completing the characterization.

Findings

The condition for Schur rigidity is both necessary and sufficient.

Schur rigid classes are precisely characterized in cominuscule varieties.

The results deepen understanding of subvariety structures in homogeneous spaces.

Abstract

Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identified a sufficient condition for a Schubert class to be Schur rigid. In this paper we show that the condition is also necessary.