It is conjectured (following the Stanley–Stembridge conjecture) that the cohomology rings of regular semisimple Hessenberg varieties yield permutation representations, but the decompositions of the modules are only known in some cases. For the Hessenberg function \(h=(h(1),n,\ldots ,n)\) , the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. We define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function \(h' = ((n-1)^{n-m},n^m)\) , and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of P-tableaux, motivated by the work of Gasharov, illustrating the connections between the \(\mathfrak {S}_n\) action on these cohomology rings and the Schur expansion of chromatic symmetric functions.
