What is the n-dimensional representation of a U(1) element?

Question

What is the $n$-dimensional matrix representation of an element $g\in U(1)$? Is it simply the $n$-dimensional identity matrix times an exponential factor $\text{e}^{\text{i}\alpha}$? This would fit the condition that its determinant is a complex number with norm one.

Answer 1

Since $U(1)$ is compact, its (continuous, complex, finite-dimensional) representations are unitary and thus the direct sum of irreps by the Peter-Weyl theorem. By the Schur lemma, such irreps are all $1$-dimensional; that is, they're given by $\chi(t) = t^n$ (identifying $U(1)$ with the unit circle in $\mathbb{C}$) for some integer $n$. The representations of $U(1)$ are thus given by $t \to (t^{n_1}, \dots, t^{n_k})$ over some basis of $\mathbb{C}^k$ for $n_i\in \mathbb{Z}$.