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Hessenberg varieties and graph theory

Hold Date 2016-11-25 15:00~2016-11-25 16:00

Place Seminar Room W1-D-610, West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Tatsuya HORIGUCHI (Osaka City University Advanced Mathematical Institute)

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.

* This seminar is combined with Topology Seminar.