Regular nilpotent partial Hessenberg varieties
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916945223221248 |
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| author | Horiguchi, Tatsuya |
| author_facet | Horiguchi, Tatsuya |
| contents | Let $G$ be a complex semisimple linear algebraic group. Fix a subset $Θ$ 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 $Θ$-ideal $I$ (which is a special lower ideal), we can define the regular nilpotent partial Hessenberg variety $\mbox{Hess}_Θ(N,I)$ in the partial flag variety $G/P$. In this manuscript we first provide a summand formula and a product formula for the Poincaré 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 $W_Θ$ generated by $Θ$. 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 $\mbox{Hess}_Θ(N,I)$ and the $W_Θ$-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 $\mbox{Hess}_Θ(N,I)$ in terms of the $W_Θ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07247 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular nilpotent partial Hessenberg varieties Horiguchi, Tatsuya Algebraic Geometry Combinatorics 14M15, 32S22, 17B22 Let $G$ be a complex semisimple linear algebraic group. Fix a subset $Θ$ 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 $Θ$-ideal $I$ (which is a special lower ideal), we can define the regular nilpotent partial Hessenberg variety $\mbox{Hess}_Θ(N,I)$ in the partial flag variety $G/P$. In this manuscript we first provide a summand formula and a product formula for the Poincaré 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 $W_Θ$ generated by $Θ$. 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 $\mbox{Hess}_Θ(N,I)$ and the $W_Θ$-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 $\mbox{Hess}_Θ(N,I)$ in terms of the $W_Θ$-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. |
| title | Regular nilpotent partial Hessenberg varieties |
| topic | Algebraic Geometry Combinatorics 14M15, 32S22, 17B22 |
| url | https://arxiv.org/abs/2405.07247 |