Hessenberg varieties are an important class of algebraic sub varieties of the flag variety. They have been shown to be related to a large number of other areas of mathematics such as representation theory and combinatorics. This project aims to explore the use of the volume polynomial and related ideas (e.g. hyperplane arrangements) to further study the geometry of Hessenberg varieties and its applications.
Impact
Thus far it has already been noticed that the cohomology of Hessenberg varieties can be described by hyperplane arrangements. Current available information about the volume polynomial suggests that its coefficients contains interesting combinatorial information so we are in the process of computing concrete examples and analyzing the resulting data. Potential applications and impact would be in the fields of Schubert calculus and combinatorics.
Student Experience
Some graduate students and postdoctoral fellows are involved in this project.
