Some characterizations of singular components of Springer fibers in the two-column case

TL;DR

This paper characterizes singular components of Springer fibers for nilpotent endomorphisms with square zero, using combinatorial, algebraic, and intersection-based criteria, and relates these to Poincaré polynomial palindromicity.

Contribution

It provides three new characterizations of singular components in the two-column case, including combinatorial, polynomial symmetry, and intersection criteria.

Findings

Singular components correspond to non-palindromic Poincaré polynomials.

A component is singular if it has many codimension-one intersections.

For general nilpotent u, singular components imply the existence of a non-palindromic Poincaré polynomial component.

Abstract

Let $u$ be a nilpotent endomorphism of a finite dimensional $\mathbb{C}$-vector space. The set ${\mathcal F}_u$ of $u$-stable complete flags is a projective algebraic variety called a Springer fiber. Its irreducible components are parameterized by a set of standard tableaux. We provide three characterizations of the singular components of ${\mathcal F}_u$ in the case $u^2=0$. First, we give the combinatorial description of standard tableaux corresponding to singular components. Second, we prove that a component is singular if and only if its Poincar\'e polynomial is not palindromic. Third, we show that a component is singular when it has too many intersections of codimension one with other components. Finally, relying on the second criterion, we infer that, for $u$ general, whenever ${\mathcal F}_u$ has a singular component, it admits a component whose Poincar\'e polynomial is not palindromic. This work relies on a previous criterion of singularity for components of ${\mathcal F}_u$ in the case $u^2=0$ by the first author and on the description of the $B$-orbit decomposition of orbital varieties of nilpotent order two by the second author.