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 wellknown Stanley–Stembridge conjecture in combinatorics to the dot action of the symmetric group Sn 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.
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Harada, Megumi; Precup, Martha; and Tymoczko, Julianna, "Toward Permutation Bases in the Equivariant Cohomology Rings of Regular Semisimple Hessenberg Varieties" (2022). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
Peer reviewed accepted manuscript.