On rational orbits in some prehomogeneous vector spaces
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908741435129856 |
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| author | Pal, Sayan |
| author_facet | Pal, Sayan |
| contents | Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group defined over $k$ and $V$ is an irreducible rational representation of $G$ with a Zariski dense open orbit. We parametrize all composition algebras over the field $k$ in terms of the orbits in some of these representations. This leads to a parametric description of the reduced Freudenthal algebras of dimensions $6$ and $9$ over $k$ (if $\text{char}(k)\neq 2,3$). We also get a parametrization for the involutions of the second kind defined on a central division $K$-algebra $B$ with center $K$, a quadratic extension of the underlying field $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On rational orbits in some prehomogeneous vector spaces Pal, Sayan Group Theory Rings and Algebras Representation Theory 17A75, 17C60, 20G05, 20G15 Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group defined over $k$ and $V$ is an irreducible rational representation of $G$ with a Zariski dense open orbit. We parametrize all composition algebras over the field $k$ in terms of the orbits in some of these representations. This leads to a parametric description of the reduced Freudenthal algebras of dimensions $6$ and $9$ over $k$ (if $\text{char}(k)\neq 2,3$). We also get a parametrization for the involutions of the second kind defined on a central division $K$-algebra $B$ with center $K$, a quadratic extension of the underlying field $k$. |
| title | On rational orbits in some prehomogeneous vector spaces |
| topic | Group Theory Rings and Algebras Representation Theory 17A75, 17C60, 20G05, 20G15 |
| url | https://arxiv.org/abs/2512.24783 |