# Finding a simple submodule of $\mathbb C[Q_8]$ that is isomorphic to a certain irrep

Given I know a certain $$2$$-dimensional irreducible representation $$V$$ of $$\mathbb C[Q_8]$$, is there a standard process to find a simple submodule of $$\mathbb C[Q_8]$$ that is isomorphic to that irreducible representation?

There's a method described here to find the four-dimensional subalgebra of $$\mathbb C[Q_8]$$ that is isomorphic to $$V^{\oplus 2}$$ but I'm looking for the simple submodule of $$\mathbb C[Q_8]$$ that is isomorphic to $$V$$, not $$V^{\oplus 2}$$.

Your 4D subalgebra $$A\subset\mathbb{C}[Q_8]$$ corresponding to the 2D irrep $$V$$ is generated by the isotypical projector
$$e=\frac{\dim V}{|G|}\sum_{g\in G}\chi_V(g^{-1})=\frac{1-\varepsilon}{2}$$
where we call $$\varepsilon$$ the order two element of $$Q_8$$. Since $$\varepsilon e=-e$$, we get
$$A=\mathrm{span}\{e,\mathbf{i}e,\mathbf{j}e,\mathbf{k}e\}\cong\mathrm{End}(V)\cong M_2\mathbb{C}.$$
Indeed, $$\mathbb{H}\otimes\mathbb{C}\to A$$ given by $$q\otimes z\mapsto qez$$ (suitably interpreted!) is an isomorphism. Here, the second factor is considered the center $$\mathbb{C}=Z(\mathbb{C}[Q_8])$$ of scalars, so $$i\in\mathbb{C}$$ and $$\mathbf{i}\in Q_8$$ are different. This isomorphism is actually standard: the elements of $$\mathbb{H}$$ can be interpreted as linear transformations of $$\mathbb{H}$$ itself (viewing $$\mathbb{H}$$ a right $$\mathbb{C}$$-vector space) by thinking of a quaternion $$q$$ as a left-multiplication-map $$L_q(x):=qx$$, and this means by picking a basis for $$\mathbb{H}$$ (e.g. $$\{1,\mathbf{j}\}$$) we can write quaternions as $$2\times 2$$ complex matrices.
The subreps of $$A$$ correspond to its ideals as an algebra, and two nice complementary ideals of $$M_2\mathbb{C}$$ correspond to the sets of matrices of the form $$[\begin{smallmatrix}\ast&0\\\ast&0\end{smallmatrix}]$$ and $$[\begin{smallmatrix}0&\ast\\0&\ast\end{smallmatrix}]$$. If we write out our aforementioned isomorphism reasonably explicitly, we can figure out basis elements for the corresponding ideals in $$A$$.