Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of $\mathrm{PGL}$ 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 linear embeddings $\iota :{\mathbb{P}}^r\hookrightarrow {\mathbb{P}}^n$ 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.

中文翻译：

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建筑物、评估拟阵和热带线性空间

仿射布鲁哈特-蒂茨建筑是提取代数群组合的几何空间。的建设$\mathrm{前列腺素酶}$参数化固定有限维向量空间中的子空间/格的标志，或者等效地，参数化固定有限维向量空间上的范数，直到相似性。Goldman 和 Iwahori 首先将其作为对称空间的分段线性模拟进行研究。半范数空间压缩了范数空间，并允许来自射影空间伯科维奇分析的自然满射限制图，该射影空间影响了自然热带化地图。受佩恩结果的启发，分析是所有热带化的极限，我们证明半范数空间是所有热带化线性嵌入的极限$\iota ：{\mathbb{磷}}^r\hookrightarrow {\mathbb{磷}}^n$并证明了紧凑线性空间的忠实热带化结果。半范数空间实际上是与通用可实现估值拟阵相关的热带线性空间。