A unified approach on Springer fibers in the hook, two-row and two-column cases

TL;DR

This paper provides a unified combinatorial and topological analysis of Springer fibers for specific nilpotent endomorphisms, focusing on hook, two-row, and two-column cases, with new characterizations and properties.

Contribution

It introduces common characterizations and algorithms for fixed flags in Springer fibers across three special cases, advancing understanding of their structure and topology.

Findings

Characterizations involving dominance relations and combinatorial algorithms

Topological properties of components and intersections deduced

Unified approach applicable to hook, two-row, and two-column cases

Abstract

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.