Given a representation $R$ of a group $G$, how can I identify which representation in the decomposition
$$ R \otimes R \otimes R = R_1 \oplus R_2 \oplus \ldots $$
are symmetric, antisymmetric or of mixed symmetry?
Some things I already know that help, but aren't sufficient:
- The highest weight of one of the symmetric representation is three times the highest weight of $R$
- The highest weight of one of the antisymmetric representation is the highest weight of $R$ plus the next highest weight of $R$ plus the third-highest weight of $R$.
- The dimension of the symmetric subspace is $dim(Sym)= \pmatrix{dim(R)+3-1 \\ 3}$ and of the antisymmetric subspace $dim(Antisym)= \pmatrix{dim(R) \\ 3}$
Using these rules helps to identify the symmetric and antisymmetric representations in the tensor product of three times a small representation like the $10$ of $SO(10)$ or the tensor product of two times a given representations. Unfortunately, for some bigger representation like the adjoint $24$ dimensional representation of $SU(5)$ these rules aren't enough to identify which representations in the decomposition
$$ 24 \otimes 24 \otimes 24 = 2*(1)+9*(24)+6*(75)+6*(126)+6*(126)+2*(175)+2*(175)+6*(200)+224+224+1000+4*(1024)+2*(1050)+2*(1050)$$
are symmetric, antisymmetric or of mixed symmetry.