# What are the irreducible representations (Clebsch-Gordan decomposition) of $\mathbf{10}\otimes \mathbf{3}$ in $SU(3)$?

Since a rank-3 tensor has 10 components and a rank-1 tensor has 3 components in $$SU(3)$$, I know that we are searching for the different irreducible representations of the tensor $$v_{ijk}w_{l}$$.

The fully symmetric part is equivalent to a rank-4 totally symmetric tensor with $$v_{(ijk}w_{l)} = x_{ijkl}$$. This will have 15 components.

But, I am not sure how to account for the rest 15. A possible setting includes a rank-2 tensor accounting for 6, rank-(1,1) accounting for 8 and one scalar using traceless property. But this is just me fixing the numbers without derivation.

$$\mathbf{10 \otimes 3 = 15 \ \oplus \ ?}$$

• You can use Young tableaux for such decompositions. Feb 7, 2020 at 18:17

In tensor notation: $$v_{(ijk)}w_l = x_{(ijkl)} + \varepsilon_{ml(k} y_{ij)}^m + \underbrace{\varepsilon_{l(jk}z_{i)}}_{0}$$ $$y^i_{(ij)} = 0$$ Or: $$\mathbf{10 \otimes 3 = 15 \ \oplus \ 15^{\prime}}$$