On the singularity of the irreducible components of a Springer fiber in ${\mathfrak{s}\mathfrak{l}_n}$

Abstract

Let ${\mathcal{B}_u}$ be the Springer fiber over a nilpotent endomorphism ${u\in {\rm End}(\mathbb{C}^n)}$. 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 the singularity of the components of ${\mathcal{B}_u}$ and show that all the components of ${\mathcal{B}_u}$ are nonsingular if and only if ${J(u)\in\{(\lambda,1,1,\ldots), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}}$.