I am trying to understand how to see if a vibrational mode is Jahn-Teller active or not. According to the group theoretical description of the Jahn-Teller effect one needs to check if the symmetric part of the direct product the irreducible representation (=irrep) of the electronic state with itself contains the irrep of the distortional mode in its symmetric part. And this is where I have trouble to understand. I can understand that one can decompose tensors (tensor products) into a symmetric and an anti-symmetric part, but I fail to understand how to apply that to the the direct product of two irreps, as they are in most cases one-dimensional. Can someone explain to me what exactly means "symmetric, anti-symmetric and non-symmetric part" in this context and how to see that in specific cases?
The symmetric and antisymmetric part of the direct product here refers to the fact that certain irreducible representations in the direct product are symmetric while others are antisymmetric (w.r.t some operations like $\sigma_v$, $C'_2$). So, for instance, when we write:
$$\Pi \times \Pi = \Sigma^+ + [\Sigma^-] + \Delta$$
the $[ \ ]$ simply means that the irreducible representation $\Sigma^-$ is the antisymmetric part of the direct product. It doesn't mean that you are somehow decomposing $\Sigma^-$ into a symmetric and antisymmetric part, and then selecting the antisymmetric one. This won't be possible since $\Sigma^-$ is a one-dimensional representation.