Cell closures for two-row Springer fibers via noncrossing matchings

Abstract

Springer fibers are a family of subvarieties of the flag variety parametrized by nilpotent matrices that are important in geometric representation theory and whose geometry encodes deep combinatorics. Two-row Springer fibers, which correspond to nilpotent matrices with two Jordan blocks, also arise in knot theory, in part because their components are indexed by noncrossing matchings. Springer fibers are paved by affines by intersection with appropriately-chosen Schubert cells. In the two-row case, we provide an elementary description of these Springer Schubert cells in terms of standard noncrossing matchings and describe closure relations explicitly. To do this, we define an operation called cutting arcs in a matching, which successively unnests arcs while ``remembering" the arc originally on top. We then prove that the boundary of the Springer Schubert cell corresponding to a matching consists of the affine subsets of cells corresponding to all ways of cutting arcs in that matching.