Springer fibers and Schubert points

TL;DR

This paper establishes a combinatorial relationship between Springer fibers and Schubert varieties, showing that for certain partitions, their Betti numbers coincide, thus linking geometric and combinatorial aspects of these varieties.

Contribution

It proves that the Betti numbers of Springer fibers for partitions with up to three rows or two columns match those of specific unions of Schubert varieties, revealing a new combinatorial connection.

Findings

Betti numbers of Springer fibers equal those of certain Schubert unions for specific partitions

Establishes a combinatorial correspondence between Springer fibers and Schubert varieties

Enhances understanding of the topology of Springer fibers in geometric representation theory

Abstract

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.