I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The 5 representation is of course just the representation of $su\left(5\right)$ by itself. After picking a Cartan subalgebra as the diagonal matrices with zero trace, we can of course see that the roots are $L_i−L_j$ where $L_i$ picks out the ith element on the diagonal, and the weights are simply $L_i$ in this case. So what is the 'weight system'?
It is supposed to be the case that I can use the weight systems of representations to show for instance that $\mathbf{5}\otimes \mathbf{5}=\mathbf{5}\oplus \mathbf{15}$.