Regular semisimple Hessenberg varieties admit actions of associated Weyl
groups on their cohomology space of each degree. In this paper, we consider the
module structure of the cohomology spaces of regular semisimple Hessenberg
varieties of type A. We define a subset of the Bialynicki-Birula basis of the
cohomology space so that they become a module generator set of the cohomology
module of each degree. We then use those generators to construct permutation
submodules of the degree two cohomology module and show that they form a
permutation module decomposition. Our construction is consistent with a known
combinatorial result by Chow on chromatic quasisymmetric functions