We know that for special orthogonal groups $SO(N)$ there exists invariant tensors (invariant under the group action). These are $\delta_{ij}$ and the totally anti-symmetric $\epsilon_{m_1,m_2,...m_N}$ tensor.

Similarly for $SU(N)$ the invariant tensors are $\delta_k^i$, $\epsilon_{m_1,m_2,...m_N}$ and $\epsilon^{m_1,m_2,...m_N}$ ($\delta_k^i$ is an invariant tensor of $U(N)$ too but not so for the $\epsilon$'s).

These objects are very useful in constructing singlets out of objects transforming under representations of $SO(N)$ or $SU(N)$.

Question 1: Are there such tensors for the Symplectic group and Exceptional groups? I am particularly interested in the groups $Sp(2N)$ and $E_7$. Is there a systematic method of obtaining the same?

Question 2. This question is just for the sake of curiosity. Can we also find invariant tensors for supergroups like $OSp(4|\mathcal{N})$ or $SU(2,2|\mathcal{N}/2)$ that appears in numerous $\mathcal{N}$-extended supersymmetric field theories?

This post imported from StackExchange Physics at 2014-09-07 07:31 (UCT), posted by SE-user Orbifold