On the singularity of the irreducible components of a Springer fiber in \({\mathfrak{s}\mathfrak{l}_n}\) (Q5962308)

Consider a nilpotent endomorphism \(u : {\mathbb C}^n \to {\mathbb C}^n\) and the associated Springer fiber \({\mathcal B}_u\) of complete flags of \({\mathbb C}^n\) that are preserved by \(u\). The nilpotent endomorphism \(u\) is determined by the sizes of its Jordan blocks and hence can be associated with a partition \(\lambda\) of \(n\). The irreducible components of \({\mathcal B}_u\) are in one-to-one corresondence with Young tableaux of shape \(\lambda\). The fundamental question of interest here is when the components are nonsingular. The main result is that all components of \({\mathcal B}_u\) are nonsingular for only four types of partitions: \((\lambda_1,1,\dots)\), \((\lambda_1,\lambda_2)\), \((\lambda_1,\lambda_2,1)\), and \((2,2,2)\). In all other cases, \({\mathcal B}_u\) has singular components. The key step in the classification is a reduction theorem. Given a component of \({\mathcal B}_u\), it corresponds (as above) to a certain Young tableaux. If the ``last'' box of the tableaux is removed, the resulting tableaux corresponds to the component of the springer fiber for a different nilpotent endomorphism. It is shown that if the resulting component is singular, than the initial one is as well. This allows the authors to partially reduce the problem to known cases. For example, from the previously known existence of singular components for the partition \((2,2,1,1)\), it follows that for all partitions with at least four Jordan blocks and at least two of length at least two, the corresponding Springer fiber has singular components. In this manner, it is shown that in all but the four special cases, singular components exist. In all but the third case special case, it was known previously that all components are nonsingular. The bulk of the paper is devoted to showing this in the third case. This uses in part the fact that the converse of the above theorem holds when the ``last'' box is in the last column. That also leads to a simpler proof for the second case.