article published Luca Battistella author Kevin Kühn author Arne Kuhrs author Martin Ulirsch author 114697 Alejandro Vargas author <jats:title>Abstract</jats:title><jats:p>Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of parameterizes 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 <jats:italic>linear</jats:italic> embeddings 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.</jats:p> Wiley2023 eng 0024-6107 1469-775010.1112/jlms.12850 1091 Battistella L, Kühn K, Kuhrs A, Ulirsch M, Vargas A. Buildings, valuated matroids, and tropical linear spaces. <i>Journal of the London Mathematical Society</i>. 2023;109(1). doi:<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a> @article{Battistella_Kühn_Kuhrs_Ulirsch_Vargas_2023, title={Buildings, valuated matroids, and tropical linear spaces}, volume={109}, DOI={<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>}, number={1e12850}, journal={Journal of the London Mathematical Society}, publisher={Wiley}, author={Battistella, Luca and Kühn, Kevin and Kuhrs, Arne and Ulirsch, Martin and Vargas, Alejandro}, year={2023} } Battistella, Luca, et al. “Buildings, Valuated Matroids, and Tropical Linear Spaces.” <i>Journal of the London Mathematical Society</i>, vol. 109, no. 1, e12850, Wiley, 2023, doi:<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>. L. Battistella, K. Kühn, A. Kuhrs, M. Ulirsch, A. Vargas, Journal of the London Mathematical Society 109 (2023). Battistella, Luca, Kevin Kühn, Arne Kuhrs, Martin Ulirsch, and Alejandro Vargas. “Buildings, Valuated Matroids, and Tropical Linear Spaces.” <i>Journal of the London Mathematical Society</i> 109, no. 1 (2023). <a href="https://doi.org/10.1112/jlms.12850">https://doi.org/10.1112/jlms.12850</a>. Battistella, L., Kühn, K., Kuhrs, A., Ulirsch, M., &#38; Vargas, A. (2023). Buildings, valuated matroids, and tropical linear spaces. <i>Journal of the London Mathematical Society</i>, <i>109</i>(1), Article e12850. <a href="https://doi.org/10.1112/jlms.12850">https://doi.org/10.1112/jlms.12850</a> L. Battistella, K. Kühn, A. Kuhrs, M. Ulirsch, and A. Vargas, “Buildings, valuated matroids, and tropical linear spaces,” <i>Journal of the London Mathematical Society</i>, vol. 109, no. 1, Art. no. e12850, 2023, doi: <a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>. 663582026-07-08T08:04:28Z2026-07-08T08:04:43Z