Let $a$ be an arbitrary element of $SL_n$. Consider next representation of $SL_n$. $T_a: SL_n \to GL_n,\;x \mapsto axa^{-1}$. I was told that this is a reducible representation. So I need to express this representation as a sum of irreducible ones. I've read that I need to find an invariant subspaces of this representation. And this is the part where I stucked.
Thanks.
We can find two invariant subspaces: $<E>$ and $A=\{M\in Mat_{n\times n} : trM=0\}$. It's easy to see that $V=<E> \oplus A$. Now we need to show that there is no proper invariant subspace in A. So we find a highest weight vector: $\begin{bmatrix}0*0 1\\ 0*00\\0*00\\0*00\end{bmatrix}$. As there is only one highest weight vector, there exist only one invariant subspace, which is our A.