# What is the weight system for these SU(5) representations?

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 $$\mathbf{5}$$ 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 $$i^{th}$$ element on the diagonal, and the weights are simply $$L_i$$ in this case.

It is supposed to be the case that I can use the weight systems of representations to show for instance that $$\mathbf{5}\otimes \mathbf{5}=\mathbf{10}\oplus \mathbf{15}$$.

The weights of an irrep $$\lambda^*$$ conjugate to $$\lambda$$ will be the negative of the weights in $$\lambda$$. The simplest way to handwave your way to this is to consider the (diagonal) transformations $$\exp\left[i\sum_k \theta_k \Lambda_k\right]\, , \tag{1}$$ where $$\Lambda_k$$ is a Cartan element. Act on a state of definite weight $$(w_1,\ldots,w_k)$$ to produce $$\exp\left[i\sum_k \theta_k \Lambda_k\right]\vert w_1\ldots w_k\rangle= \exp\left[i\sum_k \theta_k w_k\right]\vert w_1,\ldots w_k\rangle.$$ Now take the complex conjugate of (1) , which clearly produces $$\exp\left[i\sum_k \theta_k q_k\right] \vert w_1,\ldots w_k\rangle^*$$ i.e. the weights are all reversed.
This is not a very formal proof but rather an intuitive justification of the rule given about as to the relation between the weights in $$\lambda^*$$ and $$\lambda$$.
FYI the simplest way to couple copies of the fundamental irrep is to use Young tableaux. In your specific case the result must be in the $$SU(5)$$ irreps labelled by the partitions $$\{2\}$$ and $$\{1,1\}$$, corresponding to Dynkin labels $$(2,0,0,0)$$ and $$(0,1,0,0)$$. The $$(2000)$$ is fully symmetric while the $$(0100)$$ is fully antisymmetric.