Title: Valuated Matroids, Tropicalized Linear Spaces and the Affine Building of $PGL_{r+1}(K)$

Place: Seminar on Nonlinear Algebra of the Max Planck Institute for Mathematics in the Sciences

Date: 19.04.2023, 11:00

Duration: 1 hour

Abstract:

Valuated matroids were introduced by Dress and Wenzel in the 90s to combinatorially study metric spaces that arise naturally in $p$-adic geometry and in phylogenetics. In tropical geometry, they encode the information of the tropicalization of a linear space. Affine buildings were introduced by Bruhat and Tits in the 70s as highly symmetric simplicial complexes to extract the combinatorics of algebraic groups. The affine building associated to the projective linear group $PGL_{r+1}(K)$ admits a description via norms, and by work of Werner a compactification via semi-norms. Inspired by Payne’s result that the Berkovich analytification is the limit of all tropicalizations, we show that the space of seminorms on the dual of $(K^{r+2})$ is the limit of all tropicalized \emph{linear} embeddings $\iota : \mathbb{P}^r\hookrightarrow\mathbb{P}^n$ and prove a faithful tropicalization result for compactified linear spaces. Thus, under a suitable hypothesis on the non-Archimedean field $K$, the punchline is that the rank-$(r+1)$ $K$-realizable valuated matroids approximate the compactification of the affine building of $PGL_{r+1}(K)$ in a precise manner, and this can be regarded as the tropical linear space associated to a universal $K$-realizable valuated matroid. Slides/Notes: See the slides here .