I am confused about the notation.
What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ?
(1) $so(1,3)=su(2) \bigoplus su(2)$ has different representations, I don't understand why $(\frac{1}{2},\frac{1}{2})$ is irreducible, since we already have direct sum $\bigoplus$ between two $su(2)$, and direct sum usually means reducible.
The Schwartz book (page 163) says $(\frac{n}{2},\frac{n}{2})$ These are each irreducible representations of the Lorentz algebra, but reducible representations of the $su(2)$ subalgebra corresponding to spin.
(2) The group elements are acting on some space, then $\bigoplus$ means the direct sum of these spaces, like $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$. Then what does $(\frac{1}{2},\frac{1}{2})$ means, a direction product $(\frac{1}{2},0)\bigotimes (0,\frac{1}{2})$ ?
