Kronecker product recovering the initial representation?

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arivero
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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?
 

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

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