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  "body": "\n## Definition\n\nLet $G$ be a connected [[compact Lie group|9TFC6ADC1C]] and let $T$ be a [[maximal torus|3JVGZVVTB0]] of $G$.\nLet $N_G(T) = \\set{g \\in G : gTg^{-1} = T}$ be the [[normaliser|0YGN3VAA0D]] of $T$.\nThe *Weyl group* of $G$ with respect to $T$ is the [[quotient group|0KYX2N69J0]]\n$$W = W(G, T) = N_G(T)/T.$$\nIt acts on $T$ by $(nT) \\cdot t = ntn^{-1}$ for $n \\in N_G(T)$ and $t \\in T$.\n\n## Remarks\n\nExample: for $G = SU(2)$ and $T = \\set{\\operatorname{diag}(z, \\bar{z}) : \\abs{z} = 1}$, the normaliser is $T \\cup wT$ with $w = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$.\nIndeed $w \\operatorname{diag}(z, \\bar z) w^{-1} = \\operatorname{diag}(\\bar z, z)$, so $w \\in N_G(T)$, and $W$ has order $2$, acting on $T$ by $z \\mapsto \\bar z$.\nNon-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.\nThe group $W$ is finite, and it acts faithfully on $T$.\nUp to isomorphism it does not depend on $T$, since all maximal tori are conjugate.\nIt is isomorphic to the Weyl group of the root system of $G$, the group generated by the reflections in the roots.\n",
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