Representation of the circle group

Every finite-dimensional continuous complex representation of $S^1$ is unitary and decomposes into one-dimensional weight spaces. Its continuous one-dimensional characters are $z\mapsto z^m$ for $m\in \mathbb{Z}$.

For the circle group parameterized by $e^{i\theta }$ with period $2\pi$, a one-dimensional representation acts as $e^{i\theta }\mapsto e^{iQ\theta }$, and its integer label $Q$ is its charge. Periodicity forces $Q\in \mathbb{Z}$. A tensor product representation adds U(1) charges, a dual representation negates them, and restriction to the finite cyclic group generated by $e^{2\pi i/n}$ retains only $Q\mathrm{mod}n$. Integer normalization must be fixed before discussing a largest finite subgroup left by a condensate.

If a central subgroup is isomorphic to $S^1$, its weight spaces are invariant under the entire group, because every representing operator commutes with the central circle action.