Cell closures for two-row Springer fibers via noncrossing matchings
Talia Goldwasser, Meera Nadeem, Garcia Sun, Julianna Tymoczko
TL;DR
This paper provides a concrete combinatorial description of closures of Springer Schubert cells for two-row Springer fibers, where the nilpotent operator $X$ has two Jordan blocks of sizes $(n,N-n)$. It introduces standard noncrossing matchings as indexing objects and builds an explicit bijection to Springer Schubert cells via a cell-parameterization map $f_{\mathcal{M}}$, tying matchings to permutations $w_{\mathcal{M}}$ and to the paving by affines. The main result is an explicit closure formula: $\overline{\mathcal{C}_{\mathcal{M}}^X} = \bigcup_{\mathcal{A} \subseteq \mathcal{M}} cut(\mathcal{C}_{\mathcal{M}}^X, \mathcal{A})$, where cutting arcs in $\mathcal{M}$ yields new Springer Schubert cells; the proof proceeds by induction on $N$, using subspace/quotient decompositions, left-right restrictions, and fiber-bundle analyses. The framework connects geometric representation theory with explicit graph-theoretic operations (nesting/unnesting of arcs) and suggests generalizations to larger spider-web (web) diagrams for $\mathfrak{sl}_n$, highlighting deeper connections between combinatorics and the geometry of Springer fibers.
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.