---
tag: EZGTEK3HQN
kind: concept
name: Weyl group of a compact Lie group
msc: ["22E15", 22C05]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl_group
---

## Summary

Let $G$ be a compact connected Lie group with maximal torus $T$.
Its Weyl group is the finite group $N_G(T)/T$, where $N_G(T)$ is the normaliser of $T$.
It acts on $T$ by conjugation and records how elements of $T$ can be conjugate in $G$.
It is isomorphic to the Weyl group of the root system of $G$, which is generated by reflections.
It appears in the Weyl integration and character formulas.

## Explanations

### L1 human

The maximal torus is a large commutative piece of the group, but the group can still move this piece onto itself in a few ways.
The Weyl group lists these ways, up to the trivial moves inside the torus.
For the unitary group $U(n)$ with the diagonal torus, the Weyl group is the symmetric group $S_n$: one may permute the diagonal entries by conjugating with permutation matrices.
So two diagonal unitary matrices are conjugate in $U(n)$ exactly when their entries agree up to order.