The complete list of irreducible representations of $\operatorname{U}(2)$ is
\begin{align*}
\det^{\otimes m} \otimes \rho_n,
\end{align*}
where $m \in \mathbb{Z}$, $n \geq 0$, and $\det: \operatorname{U}(2) \to \mathbb{C}^\times$ is the determinant representation.