The Coxeter Flag Variety
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911411496550400 |
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| author | Bergeron, Nantel Gagnon, Lucas Spink, Hunter Tewari, Vasu |
| author_facet | Bergeron, Nantel Gagnon, Lucas Spink, Hunter Tewari, Vasu |
| contents | For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety $\operatorname{CFl}_c\subset G/B$ as the union of left-translated Richardson varieties $w^{-1}X^{wc}_w$. This is a complex of toric varieties whose geometry is governed by the lattice $\operatorname{NC}(W,c)$ of $c$-noncrossing partitions. We show that $\operatorname{CFl}_c$ is the common vanishing locus of the generalized Plücker coordinates indexed by $W\setminus\operatorname{NC}(W,c)$. We also construct an explicit affine paving of $\operatorname{CFl}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^\bullet(\operatorname{CFl}_c)$ and $H^\bullet_{T_{ad}}(\operatorname{CFl}_c)$ in terms of the induced Cayley subgraph on $\operatorname{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_23111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Coxeter Flag Variety Bergeron, Nantel Gagnon, Lucas Spink, Hunter Tewari, Vasu Algebraic Geometry Combinatorics For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety $\operatorname{CFl}_c\subset G/B$ as the union of left-translated Richardson varieties $w^{-1}X^{wc}_w$. This is a complex of toric varieties whose geometry is governed by the lattice $\operatorname{NC}(W,c)$ of $c$-noncrossing partitions. We show that $\operatorname{CFl}_c$ is the common vanishing locus of the generalized Plücker coordinates indexed by $W\setminus\operatorname{NC}(W,c)$. We also construct an explicit affine paving of $\operatorname{CFl}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^\bullet(\operatorname{CFl}_c)$ and $H^\bullet_{T_{ad}}(\operatorname{CFl}_c)$ in terms of the induced Cayley subgraph on $\operatorname{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants. |
| title | The Coxeter Flag Variety |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2601.23111 |