The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913415820214272 |
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| author | Kuehn, Kevin Kuhrs, Arne |
| author_facet | Kuehn, Kevin Kuhrs, Arne |
| contents | The Goldman-Iwahori space of seminorms on a finite-dimensional vector space over a non-Archimedean field is a non-Archimedean analogue of a symmetric space. If, in addition, $K$ is real closed, we define a signed analogue of the Goldman-Iwahori space consisting of signed seminorms. This new space can be seen as the linear algebraic version of the real analytification of projective space over $K$. We study this space with methods from real tropical geometry by constructing natural real tropicalization maps from the signed Goldman-Iwahori space to all real tropicalized linear spaces. We prove that this space is the limit of all real tropicalized linear embeddings. We give a combinatorial interpretation of this result by showing that the signed Goldman-Iwahori space is the real tropical linear space associated to the universal realizable oriented matroid. In the constant coefficient case for $K = \mathbb{R}$, we describe this space explicitly and relate it to real Bergman fans. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces Kuehn, Kevin Kuhrs, Arne Algebraic Geometry Combinatorics Geometric Topology The Goldman-Iwahori space of seminorms on a finite-dimensional vector space over a non-Archimedean field is a non-Archimedean analogue of a symmetric space. If, in addition, $K$ is real closed, we define a signed analogue of the Goldman-Iwahori space consisting of signed seminorms. This new space can be seen as the linear algebraic version of the real analytification of projective space over $K$. We study this space with methods from real tropical geometry by constructing natural real tropicalization maps from the signed Goldman-Iwahori space to all real tropicalized linear spaces. We prove that this space is the limit of all real tropicalized linear embeddings. We give a combinatorial interpretation of this result by showing that the signed Goldman-Iwahori space is the real tropical linear space associated to the universal realizable oriented matroid. In the constant coefficient case for $K = \mathbb{R}$, we describe this space explicitly and relate it to real Bergman fans. |
| title | The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces |
| topic | Algebraic Geometry Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2407.02619 |