Invariant tensors of Symplectic and Exceptional groups.

Question

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

Answer 1

Here is a partial answer: Define $Sp(2N,\mathbb{R})$ as the group of matrices $S$ such that $S \cdot\Omega\cdot S^T=\Omega$ where $\Omega_{ij}$ is a non-degenerate anti-symmetric matrix. Then $\Omega_{ij}$ is an invariant tensor similar to the Kronecker delta for orthogonal transformations. I don't think there are any more (not 100% sure).

For $E_7$: $E_7$ may be defined to be the group that preserves a antisymmetric second-rank tensor $g_{\mu\nu}$ and a totally symmetric fourth-rank tensor $f_{\mu\nu\rho\sigma}$ with $\mu,\nu,\rho,\sigma=1,2,\ldots, 56$ (i.e., the 56 dimensional representation is the defining representation). In more detail:

$$g_{\mu_1\mu_2} = {S_{\mu_1}}^{\nu_1}\ {S_{\mu_2}}^{\nu_2}\ g_{\nu_1\nu_2} \ ,$$  
with a similar one for the other tensor.

The $E_7$ description is described in P. Cvitanovic's [Group Theory][1] book. You can take a look there for the supersymmetric groups for which he has an interesting approach.

  [1]: http://www.nbi.dk/GroupTheory/

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

Comments

For the symplectic group, there are indeed no other invariants besides the trivial one. This is for example proved  at the beginning of the Chapter VI of the classical book "The classical groups: their invariants and representations"  by Weyl.

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Thanks!