Combinatorics Seminar: "On the Bia{\l}ynicki-Birula decomposition of regular semisimple Hessenberg varieties"

Abstract: Hessenberg varieties are subvarieties of the flag variety which provide a fruitful connection between geometry, representation theory of finite groups, and combinatorics. Indeed, the symmetric group acts on the cohomology of a regular semisimple Hessenberg variety, and studying this representation is related to the Stanley--Stembridge conjecture on chromatic symmetric functions. In this talk, we study a basis of the equivariant cohomology of a regular semisimple Hessenberg variety obtained by the Bia{\l}ynicki-Birula decomposition. The maximal torus acts on each cell and by analyzing this torus action we present a combinatorial description of the support of a basis element. If time permits, we will study how this provides a geometric construction of permutation module decomposition for the equivariant cohomology of permutohedral varieties. This talk is based on joint work with Soojin Cho and Jaehyun Hong.

Host: John Shareshian