# Root Systems and their Weyl Groups

Written by Amelie Strupp

#### Introduction

This project aimed at providing an interactive application to visualize Root Systems and their
corresponding Weyl groups.

The final version is available through the following link.

#### Prerequisites

This section introduces some important concepts by using visualizations generated by the application. We begin by defining the concept of a Root System:

Definition 1 (Root System [1, p. 184]). Let $$E$$ be a a finite-dimensional Euclidean vector space with the standard Euclidean inner product. A Root System $$\Phi$$ in $$E$$ is a finite set of non-zero vectors (called roots) that satisfy the following conditions:

1. The roots span $$E$$
2. The only scalar multiples of a root $$\alpha \in \Phi$$ that belong to $$\Phi$$ are $$\alpha$$ and $$-\alpha$$
3. For every root $$\alpha \in \Phi$$, the set $$\Phi$$ is closed under reflection through the hyperplane perpendicular to $$\alpha$$
4. Integrality: If $$\alpha$$ and $$\beta$$ are roots in $$\Phi$$, then the projection of $$\beta$$ onto the line through $$\alpha$$ is an integer or half-integer multiple of $$\alpha$$

While there are infinitely many root systems, they can be classified through the different types of connected Dynkin Diagrams. More precisely: there is a 1:1 correspondence between those diagrams and the indecomposable Root Systems up to isomorphism. [2, p. 6]

Two other concepts related to Root Systems are Weyl Chambers and Weyl Groups:

Definition 2 (Weyl Chambers [1, p. 210]). The complement of the set of hyperplanes orthogonal to the roots is disconnected. The individual components are called Weyl Chambers.

Definition 3 (Weyl Groups [1, p. 198]). The group of isometries of the Euclidean vector space $$E$$ generated by reflections on hyperplanes orthogonal to the roots is called the Weyl Group of the root system.

#### Our Visualization

We visualized the different possible Root Systems in up to 6 dimensions. To display objects of higher dimensionality, we employed stereographic and orthogonal projections.