University of Minnesota Combinatorics Seminar: 2021-2022

Date: 12/03/2021

Speaker: Alexander Woo

Hessenberg Varieties of Peterson/Permutohedral Type
Hessenberg varieties are certain subvarieties of the flag variety whose cohomology was recently connected to the Stanley-Stembridge conjecture about the chromatic symmetric functions of certain graphs. They have a decomposition into affine cells, giving a geometric basis for their cohomology, but unlike in more familiar examples such as Schubert varieties, the closure of a cell is not a union of cells. For a family of Hessenberg varieties which includes the toric variety for the permutahedron and the Peterson variety, we give an explicit combinatorial description of the closure of a cell. In particular, this allows us to calculate the classes of the cell closures in the cohomology ring of the flag variety. We also get some information about the singular locus of these Hessenberg varieties. This is joint work with Erik Insko (Florida Gulf Coast U.) and Martha Precup (Washington U. in St. Louis).

Hessenberg Varieties of Peterson/Permutohedral Type

Hessenberg varieties are certain subvarieties of the flag variety whose cohomology was recently connected to the Stanley-Stembridge conjecture about the chromatic symmetric functions of certain graphs. They have a decomposition into affine cells, giving a geometric basis for their cohomology, but unlike in more familiar examples such as Schubert varieties, the closure of a cell is not a union of cells.
For a family of Hessenberg varieties which includes the toric variety for the permutahedron and the Peterson variety, we give an explicit combinatorial description of the closure of a cell. In particular, this allows us to calculate the classes of the cell closures in the cohomology ring of the flag variety. We also get some information about the singular locus of these Hessenberg varieties.
This is joint work with Erik Insko (Florida Gulf Coast U.) and Martha Precup (Washington U. in St. Louis).