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

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