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 Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n}$$\end{document} on the cohomology rings H∗(Hess(S,h))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^*({{\mathcal {H}}ess}({\mathsf {S}},h))$$\end{document} 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))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^*({{\mathcal {H}}ess}({\mathsf {S}},h))$$\end{document} 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 HT∗(Hess(S,h))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^*_T({{\mathcal {H}}ess}({\mathsf {S}},h))$$\end{document} which form permutation bases for subrepresentations in HT∗(Hess(S,h))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^*_T({{\mathcal {H}}ess}({\mathsf {S}},h))$$\end{document}. 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 HT∗(Hess(S,h))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^*_T({{\mathcal {H}}ess}({\mathsf {S}},h))$$\end{document} due to Goresky, Kottwitz, and MacPherson. The constructions presented in this manuscript generalize past work of Abe–Horiguchi–Masuda, Chow, and Cho–Hong–Lee.