|タイトル||The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture|
|講演者||原田 芽ぐみ 氏（McMaster大学）|
|内容||The topic of this talk touches upon a variety of research areas including combinatorics, Lie theory, geometry, and representation theory, and I will attempt to make the talk accessible to a correspondingly wide audience, including graduate students.
The famous Stanley-Stembridge conjecture in combinatorics states that the chromatic symmetric function of the incomparability graph of a so-called (3+1)-free poset is e-positive. In this talk, we will briefly discuss this conjecture and explain how recent work of Shareshian-Wachs, Brosnan-Chow, among others, makes a rather surprising connection between this conjecture and the geometry and topology of Hessenberg varieties, together with a certain symmetric-group representation on the cohomology of Hessenberg varieties. In particular, it turns out the Stanley-Stembridge conjecture would follow if it can be proven that the cohomology of regular semisimple Hessenberg varieties (in Lie type A) are permutation representations of a certain form. I will then describe joint work with Martha Precup which proves this statement for the special case of abelian Hessenberg varieties, the definition of which is inspired by the theory of abelian ideals in a Lie algebra, as developed by Kostant and Peterson. Our proof relies on the incomparability graph of a Hessenberg function and previous combinatorial results of Stanley, Gasharov, and Shareshian-Wachs, as well as previous results on the geometry and combinatorics of Hessenberg varieties of Martha Precup.