Buildings, valuated matroids, and tropical linear spaces

Abstract

Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of $\mathrm{PGL}$ 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 $ι\colon\mathbb{P}^r\hookrightarrow\mathbb{P}^n$ 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.

Figures (3)

• Figure 1: The building $\overline{\mathcal{B}}_1(K)$ of a trivially valued field, where $\mathcal{F}$ is shorthand for "$0\subsetneq V_1\subsetneq (K^2)^{\ast}$". The coordinate $c$ is a positive real number. A norm in the homothety class corresponding to $(\mathcal{F},c)$ has generic value $1$, and value $e^{-c}$ on $V_1\setminus\{0\}$. In the case of $c=\infty$, we have a proper seminorm with kernel $V_1$.
• Figure 2: The affine Bruhat-Tits building $\overline{\mathcal{B}}_1(\mathbb{Q}_2)$
• Figure 3: The compactified cones of $\mathop{\mathrm{Trop}}\nolimits(\mathbb{P}^2, \mathop{\mathrm{id}}\nolimits)=\mathbb{T}\mathbb{P}^2$ given by flats of the uniform matroid $U_{3,3}$. This also represents the compactified apartment in the spherical building $\overline{\mathcal{B}}_2(K)$.

Theorems & Definitions (87)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1.1
• Example 1.2
• Definition 1.3
• Theorem 1.4
• Proposition 1.5
• proof
• Example 1.6