Projection of root systems and the generalized injectivity conjecture for exceptional groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909269803139072 |
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| author | Dijols, Sarah |
| author_facet | Dijols, Sarah |
| contents | Let $a$ be a real euclidean vector space of finite dimension and $Σ$ a root system in $a$ with a basis $Δ$. Let $Θ\subset Δ$ and $M = M_Θ$ be a standard Levi of a reductive group $G$ such that $a_Θ$ $= a_M / a_G$. Let us denote $d$ the dimension of $a_Θ$, i.e the cardinal of $Δ- Θ$ and $Σ_Θ$ the set of all non-trivial projections of roots in $Σ$. We obtain conditions on $Θ$ such that $Σ_Θ$ contains a root system of rank $d$. When considering the case of $Σ$ of type exceptional, we give a list of all exceptional root systems that can occur in $Σ_Θ$ and use it to prove the generalized injectivity conjecture in most exceptional groups cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18817 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projection of root systems and the generalized injectivity conjecture for exceptional groups Dijols, Sarah Representation Theory 17B22, 11F85, 20G41, 20G05 Let $a$ be a real euclidean vector space of finite dimension and $Σ$ a root system in $a$ with a basis $Δ$. Let $Θ\subset Δ$ and $M = M_Θ$ be a standard Levi of a reductive group $G$ such that $a_Θ$ $= a_M / a_G$. Let us denote $d$ the dimension of $a_Θ$, i.e the cardinal of $Δ- Θ$ and $Σ_Θ$ the set of all non-trivial projections of roots in $Σ$. We obtain conditions on $Θ$ such that $Σ_Θ$ contains a root system of rank $d$. When considering the case of $Σ$ of type exceptional, we give a list of all exceptional root systems that can occur in $Σ_Θ$ and use it to prove the generalized injectivity conjecture in most exceptional groups cases. |
| title | Projection of root systems and the generalized injectivity conjecture for exceptional groups |
| topic | Representation Theory 17B22, 11F85, 20G41, 20G05 |
| url | https://arxiv.org/abs/2407.18817 |