What is the spin-$j$ irrep of $\mathcal{U} _2\equiv U(2)$?

Question

I'm reading this paper while met a technical problem with Schur-Weyl Duality. The paper said that we can write two qubits Hilbert space, i.e. $\left(\mathbb{C}^2\right)^{\otimes 2}$ as $$\left(\mathbb{C}^2\right)^{\otimes 2} \cong\left(\mathcal{Q}_1 \otimes \mathcal{P}_{\text {trivial }}\right) \oplus\left(\mathcal{Q}_0 \otimes \mathcal{P}_{\text {sign }}\right)$$ where $\mathcal{P} _{\mathrm{sign}}$ and $\mathcal{P} _{\mathrm{trivial}}$ stands for sign and trivial irrep of the symmetric group $\mathcal{S} _2$ respectively and $\mathcal{Q} _j$ is the spin-$j$ irrep of $\mathcal{U} _2$. My problem is what is the spin-$j$ irrep of $\mathcal{U} _2$?