I would like to go in the reverse direction than the usual one. Usually, one asks how does one decompose a tensor product as a direct sum of irreducible representations, but I would like to go in the reverse direction.
I am thinking mainly of a minuscule representation $V$ of a complex semisimple Lie group $G$ restricted to a principal $SL(2,\mathbb{C})$. Thus, ultimately, $V$ is a representation of $SL(2,\mathbb{C})$, but it comes from a minuscule representation $V$ of $G$. Now here is my question: how does one write such a representation as an iterated symmetric tensor product of $\mathbb{C}^2$, when applicable, where $\mathbb{C}^2$ is the standard representation of $SL(2,\mathbb{C})$?
For example, one may consider $G = SL(5,\mathbb{C})$, with $V$ having as highest weight $e_1 + e_2$. The dimension of $V$ is thus $\binom{5}{2} = 10$. In this case, $V = \operatorname{Sym}^2(\operatorname{Sym}^3(\mathbb{C}^2))$, when restricted to a principal $SL(2,\mathbb{C})$, unless I am mistaken, which decomposes into $V \simeq \operatorname{Sym}^6(\mathbb{C}^2) \oplus \operatorname{Sym}^2(\mathbb{C}^2)$. However, in my case, I am interested in the first formula for $V$ (as an iterated symmetric tensor product of $\mathbb{C}^2$), rather than its decomposition as a direct sum of irreducible representations of $SL(2,\mathbb{C})$.
In the context I am interested in, can one always write $V$ as an iterated symmetric tensor product of $\mathbb{C}^2$? If yes, could someone perhaps provide a reference? If not, could someone perhaps indicate "the best that one can do" to write $V$ as an iterated tensor product of $\mathbb{C}^2$ (whether symmetric or not)?
I realize it may be an "unusual" question, but I will take whatever I can get. I would really appreciate it if someone could please point me to some references for what I am looking for.
Edit 1: Due to TobiasKildetoft's comment, I will rephrase my question. In particular, I will drop the requirement of $V$ being irreducible. Here is one possible reformulation. Let $\mathcal{S}$ denote the class of all complex representations of $SU(2)$ which can be written as an iterated tensor product of standard $\mathbb{C}^2$'s. One may use symmetric tensor products or the usual one, say, and possibly skew-symmetric tensor products. More generally, one may consider the subspace of the tensor product of $m$ copies of $\mathbb{C}^2$ which is invariant by a permutation group $\Gamma$ permuting the factors, and then iterate. The letter $\mathcal{S}$ is for Schur. I realize that I should try to make the definition of $\mathcal{S}$ more precise (I will think about it).
If $G$ is a compact connected semisimple Lie group, and $\rho: SU(2) \to G$ is a principal homomorphism (which is unique up to conjugation in $G$), determine all representations $V$ of $G$, which one "restricted" to a regular $SU(2)$, belong to $\mathcal{S}$.