Show that $G$ has four inequivalent 1-dimensional complex representations

representation-theory

I am looking at the exercise below in relation to representation theory

We define a group structure on $\lbrace \pm 1, \pm a, \pm b , \pm c \rbrace$ through the relations $a^2 =b^2 =c^2 =abc = 1 $ and denote the resulting group by G.

  1. Show that $G$ has four inequivalent 1-dimensional complex
    representations, and find those (Hint: $\lbrace \pm 1 \rbrace $ is a
    normal subgroup of G and $G / \lbrace \pm 1 \rbrace \simeq
    \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z} $    .)
  2. Define $\pi: G \to GL_2(\mathbb{C})$ by $\pi (\pm 1 )= \pm I_2 ,
    \pi(\pm a)=\pm \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \pi
    (\pm b)=\pm \begin{pmatrix} 0 & 1 \\
    -1 & 0 \end{pmatrix}, \pi(\pm c)= \pm \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} $    . Prove that $\pi$ is a well-defined
    irreducible
    representation of $G$.
  3. Use (1) and (2) to find the character table of G.

I have shown part 2. by using the inner product of the character, but I am having some difficulties in part 1. (and therefore also in part 3). In part 1, I am not quite sure how I am to use a normal subgroup in order to show the wanted. Additionally, I am having some difficulties in deciding what the representations explicitly are.

Best Answer

The cyclic group $\mathbb{Z}/n\mathbb{Z}$ has 1D representations $x\mapsto\zeta^x$ where $\zeta=\exp(2\pi i/n)$ is a primitive $n$th root of unity. In general, a finite abelian group has only 1D irreducible representations. If $A$ is abelian and $K$ the kernel of a representation $\rho:A\to GL_1(\mathbb{C})$, then the image is a cyclic subgroup of $S^1$ (roots of unity), and by the first isomorphism theorem, $K/A$ must be cyclic.

Given a normal subgroup $N\trianglelefteq G$ and a representation $\rho:G/N\to GL(U)$ of its quotient, there is also a projection map $\pi:G\to G/N$, and the composition $(\rho\circ\pi):G\to GL(U)$ is a rep of $G$.

In particular, for $V=\mathbb{Z}_2\times\mathbb{Z}_2$, there are three (proper nontrivial) subgroups $W$, each isomorphic to $\mathbb{Z}_2$ generated by a nontrivial element of $V$. Each of the quotients $V/W$ has a unique nontrivial representation, taking the coset $W$ to $+1$ and the other coset of $W$ to $-1$. The corresponding representation of $V$ produces $\pm1$ depending on if an element is in $W$ or not (this only works because $W$ is index $2$).

Similarly, 1D reps of $Q_8/C_2=V$ become 1D reps of $Q_8$. The three aforementioned subgroups of $V$ become the three subgroups $H=\langle\mathbf{i}\rangle,\langle\mathbf{j}\rangle,\langle\mathbf{k}\rangle$ of $Q_8$, each index $2$. Then we also have the three 1D reps of $Q_8$ defined as producing $\pm1$ (within $GL_1(\mathbb{C})$) based on whether the element of $Q_8$ is in $H$ or not. (And of course the trivial rep is 1D.)

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