Permutation Module Decomposition of the Second Cohomology of a Regular Semisimple Hessenberg Variety

Abstract Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology spaces each degree. In this paper, we consider the module structure regular type $A$. We define a subset Białynicki–Birula basis space, which becomes generator set use these generators to construct permutation submodules degree two form decomposition. Our construction is consistent with known combinatorial result by Chow chromatic quasisymmetric functions.