Let $ \Theta: G\to GL(V) $ be an irreducible complex representation of a Lie group $ G $. Then consider $ \bigoplus_\mu \Theta: G \to GL(\bigoplus_\mu V) $, the direct sum of $ \mu $ copies of $ \Theta $.
What is the manifold of all $ \Theta $ subrepresentations of $ \oplus_\mu V $? For example what is the dimension?
If $ \Theta $ is the trivial representation (or any one dimensional representation) then a $ \Theta $ subrepresentations of $ \oplus_\mu V \cong \mathbb{C}^\mu $ is just a complex line so the manifold of $ \Theta $ subrepresentations is just the complex projective space $ \mathbb{C}P^{\mu-1} $
Also if $ \mu=1 $ then the manifold of $ \Theta $ subrepresentations is just a point.
What if $ \Theta $ is not one dimensional ( and $ \mu >1 $ ). For example what if $ \Theta $ is the 2 dimensional irrep of $ S_3 $, what manifold do you get? I would imagine that whatever manifold you get would still be related to $ \mathbb{C}P^{\mu-1} $ because I would imagine picking one vector from each of the $ \mu $ copies of $ V $ and then combining will give lots of different $ \Theta $ subreps of $ \oplus_\mu V \cong \mathbb{C}^\mu $.
It's the same manifold. Denote $\mu V=\overbrace{V\oplus\cdots\oplus V}^\mu$. Any $V$-subrep of $\mu V$ is the image of a nonzero $G$-morphism, and any two such morphisms are the same up to nonzero scalars. (Schur's lemma.) Thus, the manifold you want is the quotient of $\hom_G(V,\mu V)\setminus\{0\}$ by $\mathbb{C}^\times$. But, by distributivity of hom spaces, we have $\hom_G(V,\mu V)\cong\mu \hom_G(V,V)\cong\mathbb{C}^\mu$, so he quotient is $\mathbb{CP}^{\mu-1}$.
You can generalize to isomorphic copies of any representation in any other representation using direct products of Grassmanian manifolds (exercise).