Third Year Major Oral: What do we know about nonregular Hessenberg varieties?

Abstract: A Hessenberg variety is an important subvariety of the flag variety characterized by a linear operator and a Hessenberg function. It has rich properties related to representation theory, combinatorics, and geometry, and has been studied in multiple ways. 

In 2005, Tymoczko posed several open questions about the geometric structure of Hessenberg varieties. Those questions remain open in general, since most work on Hessenberg varieties focuses on the regular case. 

In this talk, I will give an overview on what people know and don’t know about Hessenberg varieties in the nonregular case. I will present new results on nilpotent Hessenberg varieties for the minimal indecomposable Hessenberg function and answer some of Tymoczko’s open questions in this setting. I will also talk about the surprising relationship between these varieties and irreducible representations of the special linear Lie algebra.

Advisor: Martha Precup