---
tag: 363770WNM2
kind: definition
name: Weyl group of a compact Lie group
msc: ["22E15", 22C05]
defines: EZGTEK3HQN
uses: [9TFC6ADC1C, 3JVGZVVTB0, 0YGN3VAA0D, 0KYX2N69J0]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl_group
---

## Definition

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]] and let $T$ be a [[maximal torus|3JVGZVVTB0]] of $G$.
Let $N_G(T) = \set{g \in G : gTg^{-1} = T}$ be the [[normaliser|0YGN3VAA0D]] of $T$.
The *Weyl group* of $G$ with respect to $T$ is the [[quotient group|0KYX2N69J0]]
$$W = W(G, T) = N_G(T)/T.$$
It acts on $T$ by $(nT) \cdot t = ntn^{-1}$ for $n \in N_G(T)$ and $t \in T$.

## Remarks

Example: for $G = SU(2)$ and $T = \set{\operatorname{diag}(z, \bar{z}) : \abs{z} = 1}$, the normaliser is $T \cup wT$ with $w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
Indeed $w \operatorname{diag}(z, \bar z) w^{-1} = \operatorname{diag}(\bar z, z)$, so $w \in N_G(T)$, and $W$ has order $2$, acting on $T$ by $z \mapsto \bar z$.
Non-example: for the abelian group $G = \mathbb{T}^n$, the maximal torus is $G$ itself, $N_G(T) = T$, and $W$ is trivial; so $W$ does not detect anything in the abelian case.
The group $W$ is finite, and it acts faithfully on $T$.
Up to isomorphism it does not depend on $T$, since all maximal tori are conjugate.
It is isomorphic to the Weyl group of the root system of $G$, the group generated by the reflections in the roots.