Centralizer of two special unitary blocks ($C\cong U(1)\times {\mathbb{Z}}_{\gcd (m,n)}$)

For the block embedding $SU(m)\times SU(n)\subset SU(m+n)$ with $m,n\ge 2$, the centralizer is the block-scalar subgroup $C=\{\mathrm{diag}(z{I}_m,w{I}_n):|z|=|w|=1,z^m{w}^n=1\}$. 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 $(z,w)\mapsto z^a{w}^b$ 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.