Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912864528236544 |
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| author | Busch, Sira Schillewaert, Jeroen Van Maldeghem, Hendrik |
| author_facet | Busch, Sira Schillewaert, Jeroen Van Maldeghem, Hendrik |
| contents | We introduce the special and general projectivity groups attached to a simplex $F$ of a thick irreducible spherical building of simply laced type. If the residue of $F$ is irreducible, we determine the permutation group of both projectivity groups of $F$, acting on the residue of $F$ and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $\mathsf{E_6,E_7,E_8}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type Busch, Sira Schillewaert, Jeroen Van Maldeghem, Hendrik Group Theory 51E24 (Primary), 20E42 (Secondary) We introduce the special and general projectivity groups attached to a simplex $F$ of a thick irreducible spherical building of simply laced type. If the residue of $F$ is irreducible, we determine the permutation group of both projectivity groups of $F$, acting on the residue of $F$ and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $\mathsf{E_6,E_7,E_8}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide. |
| title | Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type |
| topic | Group Theory 51E24 (Primary), 20E42 (Secondary) |
| url | https://arxiv.org/abs/2407.09226 |