By Schur's Lemma, the centralizer $C(G)$ of an irreducible matrix groups $G\subseteq\mathrm{GL}(\Bbb R^d)$ is an $\Bbb R$-division algebra, and thus, isomorphic to either $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). I always had the feeling that $C(G)\cong \Bbb R$ is the generic case. This is because I only know the example $G\cong \mathrm U(1)$ (considered as a real matrix group of dimension two) with $C(G)\cong\Bbb C$. I do not know an example for $C(G)\cong\Bbb H$ at all.
Questions:
- What examples are there with $C(G)\cong \Bbb H$?
- What other examples are there with $C(G)\cong \Bbb C$?
- Have the groups with a given centralizer been "classified" in some sense? That is, is there a result of the form "all $G$ with $C(G)\cong \Bbb C$ are of the following form: ..." or the like?
Note that I want $G$ to be irreducible! But I don't care whether $G$ is finite or a Lie group or not nice at all.
Well, that is just a homework exercise I set in my representation theory class a few weeks ago. (It is based on a problem I was given myself in a class I took in Aachen last milennium.) It is giving an example for which the endomorphism ring is $\mathbb{H}$. Since I'm lazy you get the problem text:
Let $i=\sqrt{-1}$ and $G=\langle{ \left(\begin{array}{rr} 0&1\\% -1&0\\% \end{array}\right), \left(\begin{array}{rr} i&0\\ 0&-i\\% \end{array}\right)% }\rangle$ be the quaternion group of order 8.
a) Construct an irreducible representation of $G$ over the real numbers, acting on a $4$-dimensional vectorspace $V\cong \mathbb{R}^4$. ( Hint: Use an $\mathbb{R}$-basis of $\mathbb{C}$ to get an $\mathbb{R}$-basis of $\mathbb{C}^2$. To show that no 2-dimensional submodule exists, consider images of a nonzero vector $(a,b,c,d)$ in this subspace under different elements of $G$, and show that they will yield a basis of at least a 3-dimensional subspace.)
b) Determine the endomorphism ring $End_{\mathbb{R} G}(V)$. (Hint: The elements of $End_{\mathbb{R} G}(V)$ are $4\times 4$ matrices that commute with the generators of $G$. Use this to deduce conditions on their entires. Then show that every matrix fulfilling these conditions commutes with $G$.)
c) By Schur's lemma $End_{\mathbb{R} G}(V)$ must be a division ring. Can you identify it?