= Descent criterion for representations of U(n)
{title2=$q\equiv b\pmod n$}
The map $(z,U)\mapsto zU$ identifies the <unitary group> with $(U(1)\times SU(n))/K$, where $K=\{(\zeta^k,\zeta^{-k}I_n):\zeta=e^{2\pi i/n}\}$. If an irreducible <SU(n) tensor transformation> has $b$ boxes in its <Young diagram>, the central scalar $\zeta I_n$ acts as $\zeta^b$. Therefore, for integer $q$, $(z,U)\mapsto z^qR(U)$ is trivial on $K$ precisely when $\zeta^{q-b}=1$, proving the criterion. Adding a full $n$-box column changes $b$ by $n$ and leaves the condition unchanged. In particular a lift with trivial first-factor <U(1) charge> descends only if its central action is trivial; this does not prevent other lifts of the same <special unitary group> <representation> from descending.
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