Submodules of tensor representations Let $V$ be a finite dimensional vector space over a field and $T$ the tensor algebra $T=\bigoplus_{n\geq 0} T_n,$ where $T_n=V^{\otimes n}$. It's easy to see that $T$ can be viewed as a $GL(V)$-module, as $V$ is the natural module for $GL(V)$. Moreover, each $T_n$ is a $GL(V)$-submodule of $T$. This is called the $n^{th}$ tensor representation.
Can we do something analogous conversely, where we can view any $GL(V)$-module as a submodule of some $T^n$? This is surely not true for $GL(V)$ as a whole, but what about if we place certain restrictions, that is, for a finite subgroup $G\subset GL(V)$, can we say any simple $k[G]$-module is a submodule of some $T_n$?
 A: This is not true for $\text{GL}(V)$, but there is a salvage. As a $\text{GL}(V)$-representation, the decomposition of $V^{\otimes n}$ is described by Schur-Weyl duality, and in particular it is a direct sum of Schur functors $S^{\lambda}(V)$. The Schur functors applied to $V$ describe all polynomial $\text{GL}(V)$-representations (those on which $M \in \text{GL}(V)$ acts by matrices with entries polynomial in the entries of $M$). (I need some assumptions on the underlying field; I think characteristic zero suffices?) 
Non-polynomial representations include, for example, $M \mapsto |\det(M)|^r$ for real $r$ if the base field is $\mathbb{R}$ or $\mathbb{C}$; these can never occur as a subrepresentation of $V^{\otimes n}$. In other words, we really want to think about $\text{GL}(V)$ as an algebraic group here. 
For finite groups we have the following classical exercise.
Theorem: Let $G$ be a finite group and $V$ a faithful complex representation of $G$. Then every irreducible representation of $G$ occurs as a subrepresentation of $V^{\otimes n}$ for some $n$.
See, for example, this MO question. The result generalizes to an algebraically closed base field of characteristic zero and this can probably be weakened. For compact groups the above generalizes as follows.
Theorem: Let $G$ be a compact group and $V$ a faithful complex representation of $G$ (so $G$ is a Lie group). Then every irreducible representation of $G$ occurs as a subrepresentation of $V^{\otimes n} \otimes V^{\ast \otimes m}$ for some $n, m$.
See, for example, this MO question. Note also that if the image of $G$ lies in $\text{SL}(V)$, then $\Lambda^{\dim V - 1}(V) \cong V^{\ast}$ (exercise), so $V^{\ast}$ is contained in a tensor power of $V$ and we get the same result as before, with no need to take the dual. 
