Definition, Generators, and Action of the Weyl Group

Introduction

The Weyl group is a fundamental concept in Lie theory and representation theory, serving as a bridge between the algebraic structure of Lie groups and the geometric properties of root systems. Defined as the group of isometries of a root system associated with a semisimple Lie group, the Weyl group acts on the roots of the group and plays a central role in understanding the symmetries of the group. This article explores the definition of the Weyl group, its generators, and its action on the root system, emphasizing its connections to Lie theory and representation theory.

Definition of the Weyl Group

The Weyl group is a finite reflection group associated with a semisimple Lie group. Given a root system $\Phi$ of a semisimple Lie group $G$, the Weyl group $W$ is the group of automorphisms of the root system that preserve the structure of $\Phi$. These automorphisms are generated by reflections corresponding to the roots of $\Phi$. The Weyl group can be constructed as the group of isometries of the root system, which includes both reflections and translations.

The Weyl group is closely related to the symmetric group $S_n$ when the root system is the set of vectors in $\mathbb{R}^n$ with coordinates $\pm 1$. In this case, the Weyl group is the symmetric group $S_n$, and its action on the root system corresponds to permutations of the coordinates. More generally, the Weyl group acts on the root system through its reflections, preserving the structure of the root lattice and the Cartan matrix.

Generators of the Weyl Group

The Weyl group is generated by reflections corresponding to the roots of the root system. These reflections are defined by the formula:
$$ s_\alpha(w) = w \cdot \text{reflection across the hyperplane orthogonal to } \alpha, $$
where $w$ is an element of the root system and $\alpha$ is a root. The generators of the Weyl group are thus the reflections $s_\alpha$ for all roots $\alpha \in \Phi$.

The group generated by these reflections is a finite group, as the root system is finite-dimensional. The number of generators is equal to the number of roots, and the Weyl group's structure is determined by the Coxeter diagram, which encodes the relationships between the reflections. For example, in the case of the root system $A_n$, the Coxeter diagram is a straight line with $n$ nodes, corresponding to the symmetric group $S_n$.

The Weyl group acts on the root system by permuting the roots according to the reflection relations. These permutations are determined by the Cartan matrix, which encodes the angles between the root vectors. The action of the Weyl group on the root system is transitive, meaning that any two roots can be connected by a sequence of reflections, reflecting the connectedness of the root system.

Action of the Weyl Group on the Root System

The Weyl group acts on the root system $\Phi$ through its reflections, preserving the structure of the root lattice and the Cartan matrix. This action is transitive, ensuring that the root system is connected. The action can be described in terms of the following steps:

Reflections and Root Permutations: Each reflection $s_\alpha$ permutes the roots of $\Phi$ by reflecting across the hyperplane orthogonal to $\alpha$. This permutation is determined by the Cartan matrix and the specific root $\alpha$.
Root Lattice and Automorphism Group: The Weyl group acts on the root lattice $Q$ by sending each root $\alpha$ to $s_\alpha(\alpha)$. This action extends to the entire lattice, preserving the structure of the root system.
Connection to Lie Groups: The Weyl group is closely tied to the automorphism group of the root system. For example, in the case of the root system $A_n$, the Weyl group is the symmetric group $S_n$, and its automorphism group includes the group of symmetries of the root lattice.

The action of the Weyl group on the root system is crucial in understanding the representation theory of Lie groups. The Weyl group determines the decomposition of representations of the Lie group into irreducible components, as well as the structure of the Weyl algebra.

Connection to Lie Theory and Representation Theory

The Weyl group is a key component in the classification of irreducible representations of semisimple Lie groups. The representation theory of a Lie group is determined by the action of its Weyl group on the root system. Specifically, the Weyl group acts on the weight lattice of the Lie group, and the irreducible representations are classified by the conjugacy classes of the Weyl group.

In the case of finite Lie groups, such as $SU(n)$, the Weyl group is finite and acts on the root system through its reflections. The action of the Weyl group on the root system determines the structure of the representation and its characters. For example, the Weyl group acts on the weights of the Lie algebra, and the irreducible representations are indexed by the conjugacy classes of the Weyl group.

The Weyl group also plays a central role in the theory of automorphic forms and harmonic analysis on Lie groups. The action of the Weyl group on the root system is essential in understanding the symmetries of the group and its associated Lie algebras.

Conclusion

The Weyl group is a fundamental object in Lie theory, serving as a bridge between the algebraic structure of Lie groups and the geometric properties of root systems. Defined as the group of isometries of a root system, the Weyl group is generated by reflections corresponding to the roots of the system and acts on the root lattice through its reflections. Its action on the root system is transitive and determines the structure of the representation theory of semisimple Lie groups. The Weyl group's connection to the symmetric group and its role in the classification of irreducible representations highlight its significance in both algebraic and geometric contexts.

The study of the Weyl group provides deep insights into the symmetries of Lie groups and their associated algebras, making it a cornerstone of modern Lie theory. Its properties, from the structure of its generators to its action on the root system, continue to influence research in representation theory, harmonic analysis