I am trying to define the symmetry property of the spin connection $ \omega_\mu^{ab}$ using the command TableauSymmetry as follows.

```
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\alpha,\beta,\gamma,\xi}::Indices(curved, position=fixed).
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\alpha,\beta,\gamma,\xi}::Integer(0..3).
{a,b,c,d}::Indices(flat, position=independent).
{a,b,c,d}::Integer(0..3).
e^{a}_{\mu}::Vielbein.
\omega_{\mu}^{a b}::TableauSymmetry( shape={1,1}, indices{1,2} ).
```

then I try to test the symmetry of $\omega_\mu^{ab}$ (which should be antisymmetric in $(a, b)$ indices as follows.

```
testO:=\omega_{\mu}^{a b} + \omega_{\mu}^{b a};
young_project_tensor(_);
```

and this causes the kernel to crash every time. I suspect that this is caused by the fact that $\mu$ and $a$, $b$ are different types of indices. If, however, I replace $\mu$ by $c$ then everything works perfectly as expected (output is 0).

```
testO:=\omega_{a}^{b c} + \omega_{a}^{c b};
young_project_tensor(_);
```

Is there a way to deal with object carrying different types of indices like the spin connection?

Other than the spin connection, there are many other objects which transform under different symmetry groups and I would like to learn how to define them.

For example, non-Abelian gauge fields

$A_{\mu}^{a}$,

and their strengths

$F_{\mu\nu}^{a}$

can carry spacetime $\mu$ and Adjoint indices $a, b$.

In this case, how should one define them? (For $SO(n)$ groups, the adjoint is often labeled by antisymmetric pair $[a,b]$, which makes things even more complicated)

Many thanks in advance for any help.