Isomorphic $\mathbb{C}[G]$-modules

Question

So I'm reading about representation theory and I came across the following exercise:

Let $Q_8 =\{a,b|a^4=1,b^2=a^2,b^{-1}ab=a^{-1}\}$ and consider the presentation $$ρ:Q_8\rightarrow GL(2,\mathbb{C})$$ given by: $$ρ(a)=\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, ρ(b)=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ and let $V$ be the corresponding $\mathbb{C}[Q_8]$-module.
Find a submodule of $\mathbb{C}[Q_8]$ isomorphic to $V$

Now I know that two representations are equivalent if their corresponding modules are isomorphic, so if I take any matrix $A\in GL(2,\mathbb{C})$ and consider the representation $ρ'(g) = Aρ(g)A^{-1}$, then its corresponding module, say $W$, should be isomorphic to $V$.
So my three questions are the following:

1. The $W$ I will end up finding, will it be equal or isomorphic to $V$ as $\mathbb{C}$-vector spaces
2. How can I calculate a $\mathbb{C}[Q_8]$-basis for $W$
3. Is there a different way to find $W$

Thank you in advance.

Answer 1

In general, let $G$ be a finite group. Then as a $G\times G$-representation (acting from the left and right), $\mathbb C[G]:=\{\sum_{g\in G} a_g[g]:a_g\in\mathbb C\}$ decomposes as $\bigoplus_{\rho\in\mathrm{Irr}(G)}\rho\boxtimes\rho^\vee$. Explicitly, the embedding $\rho\boxtimes \rho^\vee\hookrightarrow\mathbb C[G]$ is given by $v\otimes v^\vee\mapsto \sum_{g\in G}\langle v^\vee,\rho(g^{-1})v\rangle [g]$.

In particular, if we consider $\mathbb C[G]$ as a $G$-representation just by left multiplication, then for any nonzero $v^\vee\in V^\vee$ we obtain an embedding $\iota_{v^\vee}\colon V\hookrightarrow\mathbb C[G]:v\mapsto \sum_{g\in G}\langle v^\vee,\rho(g^{-1})v\rangle [g]$.

For $G=Q_8$ and $\rho$ the prescribed representation, we can choose $v^\vee=e_1^\vee$, and $V$ is spanned by $\{\iota_{e_1^\vee}(e_1),\iota_{e_1^\vee}(e_2)\}$ (which can be calculated explicitly).