B-orbits of nilpotency order 2 and link patterns

Let B_n be the group of upper-triangular invertible n×n matrices and Xn be the set of strictly upper triangular n×n matrices of square zero seen as an algebraic variety. B_n acts on X_n by conjugation. In this paper we give first results on the geometry of orbits X_n/B_n in terms of link patterns.Further we apply this description to the computations of the closures of orbital varieties of nilpotency order 2 and their pairwise intersections. In particular, we connect our results on intersections to the combinatorics of meanders in Temperley-Lieb algebras and pairwise intersections of the components of a Springer fiber over a nilpotent element with two Jordan blocks.