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.


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