A tropical view on the Bruhat-Tits building of SL and its compactifications

TL;DR

This paper explores the structure of the Bruhat-Tits building of SL using tropical geometry, revealing how tropical Schur polynomials define compactifications of the building associated with algebraic representations.

Contribution

It introduces a novel tropical geometric approach to describe stabilizers and compactifications of the Bruhat-Tits building of SL, linking algebraic representations with tropical fans.

Findings

Stabilizers of points are described via tropical geometry.

Fans for compactification are given by tropical Schur polynomials.

Provides a new perspective on building compactifications using tropical methods.

Abstract

We describe the stabilizers of points in the Bruhat-Tits building of the group SL with tropical geometry. There are several compactifications of this building associated to algebraic representations of SL. We show that the fans used to compactify apartments in this theory are given by tropical Schur polynomials.