Shadows in the wild -- folded galleries and their applications

TL;DR

This survey explores folded galleries in Coxeter complexes and their applications in representation theory and arithmetic geometry, connecting combinatorial objects with geometric structures like affine flag varieties.

Contribution

It provides a detailed introduction to folded galleries in Coxeter complexes and explains their relevance to Bruhat-Tits buildings and affine geometric structures.

Findings

Detailed explanation of Coxeter groups and folded galleries

Connections between galleries and geometric structures in arithmetic geometry

Accessible overview for a broad mathematical audience

Abstract

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.