The geometry and combinatorics of Springer fibers

TL;DR

This survey explores the geometry and combinatorics of Springer fibers, comparing them with Schubert varieties, and discusses their relationships and open questions in geometric representation theory.

Contribution

It provides a comprehensive comparison of Springer fibers and Schubert varieties, highlighting their differences, similarities, and open problems in their geometric and combinatorial structures.

Findings

Springer fibers are key in geometric representation theory.

Schubert varieties have simpler geometric structures.

The paper discusses relationships and open questions between these varieties.

Abstract

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.