Some Characterizations of Singular Components of Springer Fibers in the Two-Column Case

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 Poincare 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 Poincare 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.