If, with a bit abuse of notation, $U\notin R\bigotimes{R}\bigotimes{R}\bigotimes{...}$ (i.e., regardless how long you clebsch up $R$, $U$ won't appear in the expansion), I call $U$ unsociable with respect to $R$ (in the group $G$). Clearly everything else is unsociable with the $1$ irrep, since (duh) $1\bigotimes{1}\bigotimes...=1$. For finite groups there can be nontrivial (ahem) cases: For $G=C\_{3v}$ (the symmetry group), $A\_2\bigotimes{A\_2}=A\_1$ and thus $E$ is unsociable w.r.t. $A\_2$.  

But what is the deal with Lie groups? E.g. is the defining irrep of $E\_7$ unsociable w.r.t. the adjoint one?



Are there "unsociable" irreps? (Definition inside)
It follows from a theorem of Burnside/Steinberg that if G is a finite group then the tensor powers of a module V contain all irreps if and ony if V is faithful for G. 


