A (very short) introduction to buildings

TL;DR

This paper provides an accessible, informal introduction to the concept of buildings in mathematics, explaining their origins, basic definitions, and examples suitable for non-experts.

Contribution

It offers a beginner-friendly, lecture-style overview of buildings, emphasizing intuition and examples over formal proofs, based on foundational work by Jacques Tits and others.

Findings

Clarifies the definition and features of buildings

Connects buildings to Coxeter groups and algebraic groups

Provides illustrative examples for non-experts

Abstract

These lectures are an informal elementary introduction to buildings. They are written for, and by, a non-expert. The aim is to get to the definition of a building and feel that it is an entirely natural thing. To maintain the lecture style examples have replaced proofs. The notes at the end indicate where these proofs can be found. Most of what we say has its origins in the work of Jacques Tits, and our account borrows heavily from the books of Abramenko and Brown and of Ronan. Lecture 1 illustrates all the essential features of a building in the context of an example, but without mentioning any building terminology. In principle anyone could read this. Lectures 2-4 firm-up and generalize these specifics: Coxeter groups appear in Lecture 2, chambers systems in Lecture 3 and the definition of a building in Lecture 4. Lecture 5 addresses where buildings come from by describing the first important example: the spherical building of an algebraic group.