An example of homogeneous cones whose basic relative invariant has maximal degree
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917666643509248 |
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| author | Nakashima, Hideto |
| author_facet | Nakashima, Hideto |
| contents | It is known that degrees of basic relative invariants of homogeneous open convex cones of rank $r$ are less than or equal to $2^{r-1}$. In this article, we show that there exists a homogeneous cone of rank $r$ one of whose basic relative invariants has degree $2^{r-1}$. The main idea for this is to construct such a homogeneous cone inductively to have specific structure constants which enable us to calculate degrees of its basic relative invariants. We study homogeneous cones of rank $3$ in detail in order to see non-triviality of the existence of homogeneous cones with given structure constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An example of homogeneous cones whose basic relative invariant has maximal degree Nakashima, Hideto Algebraic Geometry 52A20, 14P10, 22F30 It is known that degrees of basic relative invariants of homogeneous open convex cones of rank $r$ are less than or equal to $2^{r-1}$. In this article, we show that there exists a homogeneous cone of rank $r$ one of whose basic relative invariants has degree $2^{r-1}$. The main idea for this is to construct such a homogeneous cone inductively to have specific structure constants which enable us to calculate degrees of its basic relative invariants. We study homogeneous cones of rank $3$ in detail in order to see non-triviality of the existence of homogeneous cones with given structure constants. |
| title | An example of homogeneous cones whose basic relative invariant has maximal degree |
| topic | Algebraic Geometry 52A20, 14P10, 22F30 |
| url | https://arxiv.org/abs/2405.09089 |