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?
Best Answer
First I'd recomend that you use more careful language in describing the groups and representations, including the precise notion of rank. The subject is complicated enough as is, so for instance you need to refer to the Lie algebras of special orthogonal groups as having Lie type $B_n$ or $D_n$ in the respective cases when $n$ is odd, even. (Then $n$ is the rank, while the natural representation of the group has dimension $2n+1$ or $2n$.)
Your question is actually extremely difficult to answer in detail, as people understood in the era around 1900 when invariant theory was being intensively developed. Eventually, in work of Hilbert, Weyl, and others, an emphasis was developed on ring-theoretic methods here. Rather than just concentrate on the multiplicity of the trivial module in each symmetric power of a given representation, one should look at the entire graded algebra of polynomials (or polynomial functions) and try to describe the (graded) subalgebra of invariants. The starting point is the "natural" representation of the Lie algebra or Lie group, where it's already quite challenging to make the answers explicit.
Modern sources include the Springer GTM volumes 129 by Fulton-Harris and 255 by Goodman-Wallach (a second edition of an earlier book published elsewhere). Relevant work by Howe and others is covered here, along with helpful examples and references.