Linear representation theory of circle group

This article gives specific information, namely, linear representation theory, about a particular group, namely: circle group.
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This page concerns the linear representation theory of the circle group $S^{1}=\{z\in \mathbb {C} :|z|=1\}$.

The Haar integral

The map $f\mapsto {\frac {1}{2\pi }}\int _{0}^{2\pi }f(e^{i\theta })d\theta$ is a Haar integral. Thus, the representations of the circle group are all unitary, and hence completely reducible.

Thus, to understand the representations of $S^{1}$, it is enough to understand the irreducible unitary representations.

Irreducible representations

Since the group is abelian, all irreducible representations are one-dimensional.

Every irreducible representation of $S^{1}$ is of the form $z\mapsto z^{n}$ for $n$ an integer.