If I reduce the Lorentz group to the representation $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, I can write left and right-handed Weyl spinors respectively as $\left( \frac{1}{2},0 \right)$ and $\left(0, \frac{1}{2} \right)$. I can get a Dirac spinor by summing them, i.e. $\left( \frac{1}{2},0 \right) \oplus \left(0, \frac{1}{2} \right)$.
In my lecture notes, it is said that one can obtain higher representations by taking the tensor product. As an example, my prof gives $\left( \frac{1}{2},0 \right) \otimes \left(0, \frac{1}{2} \right) = \left(\frac{1}{2}, \frac{1}{2} \right)$, and he says that this corresponds to spin $1$ with $4$ components. I don't really understand what are the numbers on the right-hand side, and how I can relate this object to spin $1$. What does this notation mean, and how does the calculation work?
