Toward permutation bases in the equivariant cohomology rings of regular semisimple Hessenberg varieties

TL;DR

This paper advances the understanding of the symmetric group action on the cohomology rings of Hessenberg varieties by constructing permutation bases with stabilizers as Young subgroups, aiding in proving the Stanley-Stembridge conjecture.

Contribution

It provides new combinatorial constructions of classes in equivariant cohomology that form permutation bases with desired stabilizer properties, generalizing previous work.

Findings

Constructed permutation bases in equivariant cohomology for special Hessenberg cases.

Demonstrated stabilizers are isomorphic to Young subgroups.

Extended previous methods to broader classes of Hessenberg varieties.

Abstract

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.