On the singularity of the irreducible components of a Springer fiber in sl(n)

TL;DR

This paper investigates the singularity properties of irreducible components of Springer fibers in sl(n), establishing precise conditions under which all components are nonsingular based on the Jordan form of the nilpotent endomorphism.

Contribution

It characterizes exactly when all components of Springer fibers are nonsingular, linking singularity to specific Jordan forms of the nilpotent endomorphism.

Findings

All components are nonsingular if and only if J(u) is in a specific set of partitions.

Provides a complete classification of singular and nonsingular components based on Jordan form.

Enhances understanding of geometric structure of Springer fibers in relation to nilpotent endomorphisms.

Abstract

Let ${\mathcal B}_u$ be the Springer fiber over a nilpotent endomorphism $u\in End(\mathbb{C}^n)$. Let $J(u)$ be the Jordan form of $u$ regarded as a partition of $n$. The irreducible components of ${\mathcal B}_u$ are all of the same dimension. They are labelled by Young tableaux of shape $J(u)$. We study the question of singularity of the components of ${\mathcal B}_u$ and show that all the components of ${\mathcal B}_u$ are nonsingular if and only if $J(u)\in\{(\lambda,1,1,...), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}$.