The branching rules for $SU(8) \rightarrow USp(8)$ are: \begin{align} &8\rightarrow 8 \\ &28\rightarrow 1+27\\ &56\rightarrow 8+48\\ &70\rightarrow 1+27+42 \end{align} My question is about how vectors in the representation of $SU(8)$ change into vectors in the representation of $USp(8)$.
To be specific, for example, assume a 28-dimensional vector $F^{AB}$ with $A,B=1,...,8$, antisymmetric with $AB$, transforms in the representation 28 of $SU(8)$. Now since we know the branching rule $28\rightarrow 1+27$, in the $USp(8)$, the 28-dimensional vector $F^{AB}$ will become a 1-dimensional vector that is invariant under $USp(8)$ and a 27-dimensional vector that is in the representation 27 of $USp(8)$.
I know the 1-dimensional vector can be constructed by $F^{AB}\Omega_{AB}$, where $\Omega_{AB}$ is the symplectic invariant matrix. However, I do NOT know how to construct the 27-dimensional vector using the components of $F^{AB}$. Anyone know how to do it?
I also need to know how to deal with representation 56 and 70.
Thank you so much!