* Permutation bases for the 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
on the cohomology rings $H^*(\text{Hess}(\mathsf{S},h))$ 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 $H^*(\text{Hess}(\mathsf{S},h))$ whose elements have stabilizers isomorphic to reflection subgroups.
In this talk I will first discuss the context and background of the problem, in a manner accessible to a broad audience, and then outline several recent results which contribute to this goal.
Specifically, in some special cases, we give a new, purely combinatorial construction of classes in $H^*(\text{Hess}(\mathsf{S},h))$ which form permutation bases for subrepresentations in
$H^*(\text{Hess}(\mathsf{S},h))$. Our techniques use the Goresky-Kottwitz-MacPherson theory in equivariant cohomology. Special cases of our construction have appeared in past work of
Abe-Horiguchi-Masuda, Timothy Chow, and Cho-Hong-Lee. This is a report on joint work with Martha Precup and Julianna Tymoczko.