Codimension one intersections of the components of a Springer fiber for the two-column case

Abstract

This paper is a subsequent paper of Melnikov and Pagnon: Reducibility of the intersections of components of a Springer fiber, Indag. Mathem. 19 (4) (2008) 611–631. Here we consider the irreducible components of a Springer fibre (or orbital varieties) for the two-column case in GL n (ℂ). We describe the intersection of two irreducible components, and specially give the necessary and sufficient condition for this intersection to be of codimension one. Since an orbital variety in the two-column case is a finite union of the Borel orbits, we solve the initial question by determining orbits of codimension one in the closure of a given orbit. We show that they are parameterized by a specific set of involutions called descendants, already introduced by the first author in a previous work. Applying this result we show that the codimension one intersection of two components is irreducible and provide the combinatorial description in terms of Young tableaux of the pairs of such components.