Weyl group symmetries of the toric variety associated with Weyl chambers
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| Format: | Preprint |
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2024
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| _version_ | 1866909695219859456 |
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| author | Gui, Tao Hu, Hongsheng Liu, Minhua |
| author_facet | Gui, Tao Hu, Hongsheng Liu, Minhua |
| contents | For any crystallographic root system, let $W$ be the associated Weyl group, and let $\mathit{WP}$ be the weight polytope (also known as the $W$-permutohedron) associated with an arbitrary strongly dominant weight. The action of $W$ on $\mathit{WP}$ induces an action on the toric variety $X(\mathit{WP})$ associated with the normal fan of $\mathit{WP}$, and hence an action on the rational cohomology ring $H^*\left(X(\mathit{WP})\right)$. Let $P$ be the corresponding dominant weight polytope, which is a fundamental region of the $W$-action on $\mathit{WP}$. We give a type uniform algebraic proof that the fixed subring $H^*\left(X(\mathit{WP})\right)^{W}$ is isomorphic to the cohomology ring $H^*\left(X(P)\right)$ of the toric variety $X(P)$ associated with the normal fan of $P$. Notably, our proof applies to all finite (not necessarily crystallographic) Coxeter groups, answering a question of Horiguchi--Masuda--Shareshian--Song about non-crystallographic root systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_13617 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weyl group symmetries of the toric variety associated with Weyl chambers Gui, Tao Hu, Hongsheng Liu, Minhua Algebraic Topology Primary 13A50, Secondary 14M25, 17B22, 52B05 For any crystallographic root system, let $W$ be the associated Weyl group, and let $\mathit{WP}$ be the weight polytope (also known as the $W$-permutohedron) associated with an arbitrary strongly dominant weight. The action of $W$ on $\mathit{WP}$ induces an action on the toric variety $X(\mathit{WP})$ associated with the normal fan of $\mathit{WP}$, and hence an action on the rational cohomology ring $H^*\left(X(\mathit{WP})\right)$. Let $P$ be the corresponding dominant weight polytope, which is a fundamental region of the $W$-action on $\mathit{WP}$. We give a type uniform algebraic proof that the fixed subring $H^*\left(X(\mathit{WP})\right)^{W}$ is isomorphic to the cohomology ring $H^*\left(X(P)\right)$ of the toric variety $X(P)$ associated with the normal fan of $P$. Notably, our proof applies to all finite (not necessarily crystallographic) Coxeter groups, answering a question of Horiguchi--Masuda--Shareshian--Song about non-crystallographic root systems. |
| title | Weyl group symmetries of the toric variety associated with Weyl chambers |
| topic | Algebraic Topology Primary 13A50, Secondary 14M25, 17B22, 52B05 |
| url | https://arxiv.org/abs/2410.13617 |