# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1701.03502