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  "body": "\n## Summary\n\nLet $G$ be a compact connected Lie group with maximal torus $T$.\nIts Weyl group is the finite group $N_G(T)/T$, where $N_G(T)$ is the normaliser of $T$.\nIt acts on $T$ by conjugation and records how elements of $T$ can be conjugate in $G$.\nIt is isomorphic to the Weyl group of the root system of $G$, which is generated by reflections.\nIt appears in the Weyl integration and character formulas.\n\n## Explanations\n\n### L1 human\n\nThe maximal torus is a large commutative piece of the group, but the group can still move this piece onto itself in a few ways.\nThe Weyl group lists these ways, up to the trivial moves inside the torus.\nFor 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.\nSo two diagonal unitary matrices are conjugate in $U(n)$ exactly when their entries agree up to order.\n",
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