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We consider bases for the cohomology space of regular semisimple Hessenbergvarieties, consisting of the classes that naturally arise from theBialynicki-Birula decomposition of the Hessenberg varieties. We give anexplicit combinatorial description of the support of each class, which enablesus to compute the symmetric group actions on the classes in our bases. We thensuccessfully apply the results to the permutohedral varieties to explicitlywrite down each class and to construct permutation submodules that constitutesummands of a decomposition of cohomology space of each degree. This resolvesthe problem posed by Stembridge on the geometric construction of permutationmodule decomposition and also the conjecture posed by Chow on the constructionof bases for the equivariant cohomology spaces of permutohedral varieties.

Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties