Background (skip if you are not interested):
I recently began my study of the group representation theory. I encountered this exercise:
Find all irreducible representations of the quaternion group $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$ over $\mathbb{C}$.
Suppose $n_i$ are the dimensions of all irreducible representations. Then $\sum_i n_i^2=8$. One dimensional representations can be found by brutal force: Since $\mathrm{Aut}_{\mathbb{C}}(\mathbb C)=\mathbb C^\times$ is commutative, a representation $(\rho,\mathbb C)$ factors through its abelianization $Q_8/\{\pm1\}$, and hence we check that:
- $\rho(i)=1,\ \rho(j)=1$
- $\rho(i)=1,\ \rho(j)=-1$
- $\rho(i)=-1,\ \rho(j)=1$
- $\rho(i)=-1,\ \rho(j)=-1$
are the only four distinct 1-dimensional representations. Hence $n_1,n_2,n_3,n_4=1$ and $n_5\geq2$. Recall that $\sum_i n_i^2=8$ Hence $n_5=2$ and they exhaust all irreducible representations.Finally to find the unique 2-dimensonal irreducible representation, consider
$$\rho:Q_8\to GL_2(\mathbb C)$$ defined by ($I=$ unit matrix) $$\pm1\mapsto\pm I,\quad i\mapsto\left(\matrix{\sqrt{-1}&\\&1}\right),\quad j\mapsto\left(\matrix{1&\\&\sqrt{-1}}\right),\quad k\mapsto\left(\matrix{\sqrt{-1}&\\&\sqrt{-1}}\right)$$ We check that this indeed a well-defined group homomorphism. It remains to show $(\rho,\mathbb C^2)$ is irreducible. Indeed, if there is a subrepresentation it must be some 1-dimensional $V\subset\mathbb C^2$ invariant under $G$. This means the action of $G$ restricted to $V$ has order $2$, i.e. $-I$ must be trivial on $V$, which is absurd unless $V=0$.
Actually I just realized that the construction above is wrong. Now I want to find a paradigm even more!
Questions:
While browsing through similar questions I found this on MO. It says that the 2-dimensional representation can be obtained from the left action of $Q_8$ on the real quaternions (viewed as a 4-dimensional real vector space). I am not sure I understand. How does this method work?
Is it a paradigm to consider the action of a group on its group algebra over some field to obtain its irreducible representation? If not, is there some procedure I can follow when I meet a different group?
For the quaternion group, its irr. reps. over $\mathbb R$ and over $\mathbb C$ are (almost) identical. Does this mean there is a link between reps. over $\mathbb R$ and $\mathbb C$. More specifically, if we have obtained irr. reps. over $\mathbb R$, can we use the infomation to find the ones over $\mathbb C$ or vice versa?
I did this exercise because I need it to draw the character table of $Q_8$. One would natually think that there is some infomation lost if the character table is all we have. Does this mean we can somehow find the character table without listing all irreducible representations of a group?
p.s. I am only a beginner. So a method that doesn't require introducing a deeper theory is preferred.