Regular nilpotent partial Hessenberg varieties

TL;DR

This paper studies the cohomology rings of regular nilpotent partial Hessenberg varieties, providing formulas for their Poincaré polynomials and generalizing known invariance results to these varieties.

Contribution

It introduces summand and product formulas for Poincaré polynomials and extends the invariance of cohomology rings under parabolic Weyl group actions to partial Hessenberg varieties.

Findings

Derived explicit formulas for Poincaré polynomials.

Established isomorphisms between cohomology rings and invariant subrings.

Connected the cohomology of Hessenberg varieties to logarithmic derivation modules.

Abstract

Let $G$ be a complex semisimple linear algebraic group. Fix a subset $\Theta$ of simple roots. Given a lower ideal $I$ in positive roots, one can define the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$ in the full flag variety $G/B$. For a $\Theta$-ideal $I$ (which is a special lower ideal), we can define the regular nilpotent partial Hessenberg variety $\mbox{Hess}_\Theta(N,I)$ in the partial flag variety $G/P$. In this manuscript we first provide a summand formula and a product formula for the Poincar\'e polynomial of regular nilpotent partial Hessenberg varieties. It is a well-known result from Bernstein-Gelfand-Gelfand that the cohomology ring of the partial flag variety $G/P$ is isomorphic to the invariants in the cohomology ring of the full flag variety $G/B$ under an action of the parabolic Weyl group $W_\Theta$ generated by $\Theta$. We generalize this result to regular nilpotent partial Hessenberg varieties. More concretely, we give an isomorphism between the cohomology ring of a regular nilpotent partial Hessenberg variety $\mbox{Hess}_\Theta(N,I)$ and the $W_\Theta$-invariant subring of the cohomology ring of the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$. Furthermore, we provide a description of the cohomology ring for a regular nilpotent partial Hessenberg variety $\mbox{Hess}_\Theta(N,I)$ in terms of the $W_\Theta$-invariants in the logarithmic derivation module of the ideal arrangement $\mathcal{A}_I$, which is a generalization of the result by Abe-Masuda-Murai-Sato with the author.