2009/09/22 by Fresse, Lucas, Melnikov, Anna
#05E10 #14M15 (primary) #20C08 (secondary) #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.0909.4008
Let u be a nilpotent endomorphism of a finite dimensional ℂ-vector space. The set \mathcal Fu 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 Fu in the case u2=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 \mathcal Fu 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 \mathcal Fu in the case u2=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.