This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group.
I was thinking about its manifestation in group representations. We might as well look at a finite group $G$ and its classes of irreducible representations $\hat{G}$, and some possible candidates might be \begin{equation} d_{\alpha}=\operatorname{degree}(\alpha) \end{equation} and \begin{equation} n_{d}=\#\{\alpha\in\hat{G}:d_\alpha=d\}. \end{equation}Some evidence might be contained in
$d_{\alpha}$ divides $\#G/Z(G)$ for all $\alpha\in\hat{G}$.
So the larger $Z(G)$ is, the more commutative $G$ is, the smaller $d_\alpha$ is. On the other hand,
$n_1=\#G/[G,G]$.
So the larger $[G,G]$ is, the less commutative $G$ is, the smaller $n_1$ is.
So, I am wondering whether these can be made precise. And are there other manifestations of the $Z(G)/[G,G]$-duality in group representations?
Thanks very much!
