Rational orbits in some prehomogeneous vector spaces associated to $Sp_{6}$ revisited
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908741441421312 |
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| author | Pal, Sayan |
| author_facet | Pal, Sayan |
| contents | Let $k$ be a field with $\text{char}(k)\neq 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $Sp_{6}$-orbits in a Zariski-dense $Sp_{6}$-invariant subset $V^{ss}\subset V=\wedge^{3}V_{6}$, where $V_{6}$ is the standard $6$-dimensional irreducible representation of $Sp_{6}$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ and $(GSp_{6}\times GL_{1}^{2},V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational orbits in some prehomogeneous vector spaces associated to $Sp_{6}$ revisited Pal, Sayan Group Theory Rings and Algebras Representation Theory 17A75, 17C60, 20G05, 20G15 Let $k$ be a field with $\text{char}(k)\neq 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $Sp_{6}$-orbits in a Zariski-dense $Sp_{6}$-invariant subset $V^{ss}\subset V=\wedge^{3}V_{6}$, where $V_{6}$ is the standard $6$-dimensional irreducible representation of $Sp_{6}$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ and $(GSp_{6}\times GL_{1}^{2},V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces. |
| title | Rational orbits in some prehomogeneous vector spaces associated to $Sp_{6}$ revisited |
| topic | Group Theory Rings and Algebras Representation Theory 17A75, 17C60, 20G05, 20G15 |
| url | https://arxiv.org/abs/2512.24789 |