Maximal tori and Weyl groups of U(n) and SU(n)

The diagonal unitary matrices form a compact connected abelian subgroup of $U(n)$. A matrix commuting with every diagonal unitary matrix has zero $(i,j)$-entry for $i\mathrm{\ne }j$, by choosing diagonal phases whose $i$th and $j$th entries differ; hence the centralizer of $T_{U(n)}$ is itself, so it is maximal. For $n\ge 2$ the same entrywise argument uses determinant-one diagonal phases and shows that the centralizer of $T_{SU(n)}$ in $SU(n)$ is $T_{SU(n)}$; for $n=1$, $SU(1)$ is trivial. Thus both displayed tori are maximal.

The permutation matrices ${\pi }_{\sigma }$ are unitary and satisfy ${\pi }_{\sigma }\mathrm{diag}(z_1,\dots ,z_n){\pi }_{\sigma }^{-1}=\mathrm{diag}(z_{\sigma (1)},\dots ,z_{\sigma (n)})$, so they lie in the normalizer and induce $S_n$ in the Weyl group. For $SU(n)$ choose a diagonal unitary $d_{\sigma }$ with $\det d_{\sigma }=(\det {\pi }_{\sigma }{)}^{-1}$; then $d_{\sigma }{\pi }_{\sigma }\in SU(n)$ and, because $d_{\sigma }$ commutes with the diagonal torus, it induces the same coordinate permutation.

Conversely, let $g$ normalize the diagonal torus. Choose a regular element $t=\mathrm{diag}(z_1,\dots ,z_n)$ of the torus with pairwise distinct $z_j$. Then $gt{g}^{-1}\in T$, and the $z_j$ are the eigenvalues of $t$; since $gt{g}^{-1}$ has the same eigenvalues, and the eigenspaces of $t$ are the coordinate lines, $g$ permutes those lines up to scalars, hence equals a permutation matrix times a diagonal matrix. In $U(n)$ this says that the normalizer is generated by the torus and the permutation matrices. In $SU(n)$, if the induced permutation is $\sigma$, step 2.1 supplies the determinant-corrected representative $d_{\sigma }{\pi }_{\sigma }\in SU(n)$; multiplying $g$ by its inverse leaves a diagonal determinant-one matrix, so the normalizer is generated by $T_{SU(n)}$ and these corrected representatives. In either case the quotient is $S_n$.

Consequently the Weyl groups of $U(n)$ and $SU(n)$ are $S_n$ acting by coordinate permutation, in agreement with the root-system computation for types $A_{n-1}$.