Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators

TL;DR

This paper investigates the geometric structure of varieties related to invariant subspaces of nilpotent operators, focusing on orbit dimensions and degenerations, with implications for understanding the algebraic and combinatorial properties of these spaces.

Contribution

It provides a detailed analysis of orbit dimensions and a combinatorial characterization of degenerations in varieties of invariant subspaces, advancing the understanding of their geometric and algebraic structure.

Findings

Dimensions of orbits under group actions are explicitly computed.

A combinatorial description of the partial order of degenerations is established.

The study enhances understanding of the geometric properties of invariant subspace varieties.

Abstract

We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.