Toward Permutation Bases in the Equivariant Cohomology Rings of Regular Semisimple Hessenberg Varieties

Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the well-known Stanley–Stembridge conjecture in combinatorics to the dot action of the symmetric group S_n on the cohomology rings H^∗(Hess(S,h)) of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley–Stembridge conjecture, it suffices to construct (for any Hessenberg function h) a permutation basis of H^∗(Hess(S,h)) whose elements have stabilizers isomorphic to Young subgroups. In this manuscript, we give several results which contribute toward this goal. Specifically, in some special cases, we give a new, purely combinatorial construction of classes in the T-equivariant cohomology ring H_T^∗(Hess(S,h)) which form permutation bases for subrepresentations in H_T^∗(Hess(S,h)). Moreover, from the definition of our classes it follows that the stabilizers are isomorphic to Young subgroups. Our constructions use a presentation of the T-equivariant cohomology rings H_T^∗(Hess(S,h)) due to Goresky, Kottwitz, and MacPherson. The constructions presented in this manuscript generalize past work of Abe–Horiguchi–Masuda, Chow, and Cho–Hong–Lee.