Representations of SU(N)

Question

I'm reading this paper by Wittten. On page 374, 2nd paragraph, he states that $$ R\otimes R =\oplus_{i=1}^sE_i $$ where $E_i$ is a distinct irreducible representation. He then gives a special case for the group $SU(N)$ and states that if $R$ is the defining $N$ dimensional representation then $s=2$. It isn't clear to me why $s=2$; perhaps if someone could tell me what the 'defining' $N$ dimensional representation was I could work out an example.

Answer 1

By definition $SU(N)$ has a representation $R$ on $\Bbb C^n$ where a matrix $M$ acts on a vector $v$ by multiplication giving $Mv$. I call $R$ the tautological representation. As our group is $SU(N)$ the representation $R\otimes R$ is isomorphic to the set of $N$-by-$N$ matrices where $SU(N)$ acts by conjugation $(M,A)\mapsto MAM^{-1}$. This breaks up as the direct sum of two irreducible representations: the scalar matrices, and the trace-zero matrices.