## Abstract

Let u be a nilpotent endomorphism of a finite dimensional ℂ-vector space. The set 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 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 Poincaré 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 F _{u} has a singular component, it admits a component whose Poincaré polynomial is not palindromic. This work relies on a previous criterion of singularity for components of 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.

Original language | English |
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Pages (from-to) | 1063-1086 |

Number of pages | 24 |

Journal | Algebras and Representation Theory |

Volume | 14 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2011 |

## Keywords

- Flag varieties
- Link patterns
- Poincaré polynomial
- Singularity criteria
- Springer fibers
- Young tableaux

## ASJC Scopus subject areas

- General Mathematics