A Decomposition of the Set of Subspaces Stable Under a Nilpotent Transformation

Abstract

For a nilpotent linear transformation f: V → V of Jordan type λ, let S(T) be the set of f-stable subspaces W of V associated with an LR (Littlewood–Richardson) tableau T. We show that (i) S(T) is a disjoint union of locally closed sets parametrized by LR-fillings of T, each of which is birationally isomorphic to an affine space, and that (ii) a~conjecture of Shimomura on semistandard d-tableaux in a rectangular diagram is true.

Key Words:

• Grassmannian
• Jordan types
• Littlewood–Richardson tableaux

2010 Mathematics Subject Classification:

• Primary: 05E10
• Secondary: 14M15

Notes

Communicated by K. Misra.