I'm taking a course on representation theory this semester, and the professor assumed we had some knowledge in central extensions. Unfortunately, I'm not very familiar with the topic, and I'm having some trouble really grasping it. I'm not entirely sure what the use of such an object is, or what the relationship between the center and the central extension is.
On our homework, we have a few questions related to finding irreducible representations of the central extension of the alternating group of order $5$, $A_5$. I've asked the professor about how to find this central extension and its representations, and he's explained it to me, but I don't fully understand what exactly he is doing. Effectively, he says the central extension $\tilde A$ is a short exact sequence of the form
$$ 1 \rightarrow \mathbb{Z}_{2} \rightarrow \tilde A \rightarrow A_5 \rightarrow 1 $$
which is in some way related to the central extension of $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ as $Q_{8}$, the quternion group in the short exact sequence:
$$ 1 \rightarrow \mathbb{Z}_{2} \rightarrow Q_{8} \rightarrow \mathbb{Z}_{2} \times \mathbb{Z}_{2} \rightarrow 1 $$
I'm not sure how these two short exact sequences are related. Can anyone explain what the relationship between these two central extensions is? Further, my professor claims that since $A_5$ is embedded in $PGL_{2}(\mathbb{C})$ the two dimensional projective linear group, we have an embedding of $\tilde A$ in $PG_{2}(\mathbb{C})$, the general linear group, and through this embedding we can determine two irreducible representations. I really don't see how this works at all. How can we find this second embedding, and how can we derive these two irreducible representations from it?
Best Answer
In general, extending a group $G$ by a group $H$ is a short exact sequence $1\to H\to\tilde G\to G\to 1$. The easiest example is $1\to H\to H\times G\to G\to 1$, often called a trivial extension, but in general there are more interesting extensions. Both $\tilde A_5$ and $Q_8$ are examples of non-trivial extensions (of $A_5$ and $\mathbb Z/2\times\mathbb Z/2$, respectively).
What your professor is showing is a general construction of how to extend a group $G\subset\mathrm{PGL}_2(\mathbb C)$ by $\mathbb Z/2$. Let $\pi\colon\mathrm{SL}_2(\mathbb C)\to\mathrm{PGL}_2(\mathbb C)$ be the obvious map, with kernel $\mathbb Z/2$. Then we obtain a short exact sequence $1\to\mathbb Z/2\to\tilde G:=\pi^{-1}(G)\to G\to 1$.
For example, applying this construction to $$G=\left\{\begin{pmatrix}1\\&1\end{pmatrix},\begin{pmatrix}1\\&-1\end{pmatrix},\begin{pmatrix}&1\\1&\end{pmatrix},\begin{pmatrix}&1\\-1\end{pmatrix}\right\}$$ gives the extension $$\tilde G=\left\{\begin{pmatrix}\pm1\\&\pm1\end{pmatrix},\begin{pmatrix}\pm i\\&\mp i\end{pmatrix},\begin{pmatrix}&\pm i\\\pm i&\end{pmatrix},\begin{pmatrix}&\pm1\\\mp1\end{pmatrix}\right\},$$ which is the quaternion group $Q_8$. Similarly, there is an exceptional isomorphism $\mathrm{PGL}_2(\mathbb C)\cong\mathrm{SO}_3(\mathbb C)$, and thus $A_5\subset\mathrm{SO}_3(\mathbb R)$, the rotational symmetry group of the dodecahedron, can be viewed as a subgroup of $\mathrm{PGL}_2(\mathbb C)$. The above construction then gives an extension $\tilde A_5$ of $A_5$ by $\mathbb Z/2$.