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Multi-rigidity of Schubert classes in partial flag varieties

Yuxiang Liu, Artan Sheshmani, Shing-Tung Yau

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.

Multi-rigidity of Schubert classes in partial flag varieties

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 , we deduce the rigidity and multi-rigidity from the corresponding generalized Grassmannians (correspond to maximal parabolics). When is semi-simple, we also deduce the rigidity and multi-rigidity from the simple cases.

Paper Structure

This paper contains 18 sections, 52 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Partial flag varieties of type A
  3. Orthogonal Grassmannians
  4. Orthogonal partial flag varieties
  5. General homogeneous varieties
  6. Preliminaries
  7. Root systems of algebraic groups
  8. Partial flag varieties (Type $A$)
  9. Odd orthogonal partial flag varieties (Type $B$)
  10. Even orthogonal partial flag varieties (Type $D$)
  11. Partial flag varieties of type A
  12. Orthogonal partial flag varieties
  13. Compatibility between different components
  14. Restriction varieties
  15. Multi-rigidity of sub-indices
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $\sigma_{a^\alpha}$ be a Schubert class of $F(d_1,...,d_k;n)$. An essential $a_i$ is multi-rigid if and only if $a_i-a_{i-1}=1$ and one of the following conditions hold:

Theorems & Definitions (110)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 100 more