Kronecker product recovering the initial representation?

[arivero]

In E6, the product 27 x 27 contains the (conjugate) 27. In SU(3), something similar happens with 3 x 3, which decomposes as 3 + 6.

I was wondering, how usual is this? Do we have some lemmas telling when a product N x N is going to "recover" the original N, or its conjugate, inside the sum?

 

[arivero]

so, what happens is that R x R always decompose in a sum of the symmetric plus alternating (or exterior) square.