Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lambda$, indexed by its highest weight $\lambda = (\lambda_1, \dots, \lambda_n)$, what is known about the decomposition of the symmetric powers of $\Gamma_\lambda$?

To be more specific, suppose I am considering a Lie algebra inside $\mathfrak{gl}({n^2})$, and I have a representation $(\rho, \Gamma_\lambda)$ of one of the above two Lie algebras of equal rank to that of $\mathfrak{gl}(n^2)$ i.e., of rank $n^2$. My interest is specifically concerned with the occurrence of the trivial representation in the $n$th symmetric power of such a representations i.e., $\mathrm{Sym}^n \circ \rho$.

As a concrete example, consider the rank 9 representation of $\mathfrak{so}_4$ with highest weight $8 \omega_1$ where $\omega_i$ denotes the $i$th fundamental weight. Upon applying $\mathrm{Sym}^3$ to $\Gamma_{8\omega_1}$, this representation decomposes as a direct sum of subrepresentations, one of which is the trivial representation (I computed this with Sage).

Is there any literature on the occurrence of the trival representation inside $\mathrm{Sym}^n$ powers of such representations as above?