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?