Buildings, valuated matroids, and tropical linear spaces
- Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings ι:Pr↪Pn and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.
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| Author: | Luca BattistellaORCiDGND, Kevin Kühn, Arne Kuhrs, Martin UlirschORCiDGND, Alejandro José Vargas De LeónORCiDGND |
|---|---|
| URN: | urn:nbn:de:hebis:30:3-823628 |
| DOI: | https://doi.org/10.1112/jlms.12850 |
| ISSN: | 1469-7750 |
| ISSN: | 0024-6107 |
| ArXiv Id: | http://arxiv.org/abs/2304.09146 |
| Parent Title (English): | Journal of the London Mathematical Society |
| Publisher: | Wiley |
| Place of publication: | Oxford |
| Document Type: | Article |
| Language: | English |
| Date of Publication (online): | 2023/12/28 |
| Date of first Publication: | 2023/12/28 |
| Publishing Institution: | Universitätsbibliothek Johann Christian Senckenberg |
| Release Date: | 2024/02/23 |
| Volume: | 109.2024 |
| Issue: | e12850 |
| Article Number: | e12850 |
| Page Number: | 37 |
| Institutes: | Informatik und Mathematik / Mathematik |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| MSC-Classification: | 14-XX ALGEBRAIC GEOMETRY / 14Txx Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] |
| Sammlungen: | Universitätspublikationen |
| Licence (German): |  Creative Commons - Namensnennung 4.0 |
