分野別セミナー

Hessenberg varieties are subvarieties of a flag variety. The study of the topology of Hessenberg varieties makes connections with many research areas such as: geometric representation theory, quantum cohomology of a flag variety, graph theory, and hyperplane arrangements. We talk about Hessenberg varieties divided into a first half and a second half as follows:

In the first half, we consider a Hessenberg variety in type A and it is determined by two data; a linear operator and a Hessenberg function. When the linear operator is regular nilpotent (resp. regular semisimple), the corresponding Hessenberg variety is called regular nilpotent Hessenberg variety (resp. regular semisimple Hessenberg variety). In this talk, I talk about an explicit presentation of the cohomology rings of regular nilpotent Hessenberg varieties, and an interesting connection between the cohomology rings of regular nilpotent Hessenberg varieties and the cohomology rings of regular semisimple Hessenberg varieties. Moreover, as a relation with this result, I also talk about Shareshian and Wachs conjecture which is a beautiful relationship between an representation of a symmetric group on the cohomology ring of the regular semisimple Hessenberg variety and the chromatic quasi-symmetric function of some graph. This is joint work with Hiraku Abe, Megumi Harada, and Mikiya Masuda.



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