= Centralizer of two special unitary blocks
{title2=$C\cong U(1)\times\mathbb Z_{\gcd(m,n)}$}
For the block embedding $SU(m)\times SU(n)\subset SU(m+n)$ with $m,n\geq2$, the <centralizer> is the block-scalar <subgroup> $C=\{\operatorname{diag}(zI_m,wI_n):|z|=|w|=1,\ z^mw^n=1\}$. <Schur lemma> makes the diagonal blocks scalar and kills the off-diagonal <intertwiners>, since the two blocks are inequivalent <representations> of the product <group>.
Put $d=\gcd(m,n)$, $a=m/d$, $b=n/d$. The map $(z,w)\mapsto z^aw^b$ has image the $d$th <roots of unity> and connected <kernel> $(z,w)=(e^{ib\theta},e^{-ia\theta})$. A <Bezout identity> completes $(a,b)$ to an invertible integer coordinate change on the two-dimensional torus, proving $C\cong U(1)\times\mathbb Z_d$. Thus the full <centralizer> is a <circle group> exactly when $m,n$ are coprime. For $m=n=2$, $(zw)^2=1$ gives two circles, a useful counterexample to the unrestricted connectedness claim.
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