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This is a follow-up to

Conceptual reason why Coxeter groups are never simple

A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is a matrix in $ U_n $ with $ n-1 $ dimensional $ \lambda=1 $ eigenspace. Real complex reflection groups are Coxeter groups.

I was looking at the complex reflection groups given here

https://en.wikipedia.org/wiki/Complex_reflection_group

and it seems that none of them are perfect (although the $ S_n $ and a number of the exceptional cases have index 2 perfect subgroups).

Is it true that complex reflection groups are never perfect? Is there a nice conceptual proof of this fact?

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Complex reflections have the form $$\begin{pmatrix}\zeta &0&\cdots& 0\\ 0&1&\cdots &0 \\ \vdots &&\ddots & \\ 0&\cdots &0&1\end{pmatrix}$$ and so do not have determinant $1$. Therefore the subgroup is not a subgroup of $\mathrm{SL}_n(\mathbb{C})$, and the determinant map has a kernel that is not the whole group.

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  • $\begingroup$ oh lol its so obvious in retrospect... so the abelianization is always at least size $ n_{max} $ where $ n_{max} $ is the largest of the $ n $ showing up in the $ \zeta_n $ eigenvalues of the pseusoreflections. I guess this is the complex version of the fact $ A_n $ is the determinant 1 subgroup of the permutation matrices i.e. sign of a permutation. It's cool to think of the determinant of a unitary matrix as a sort of generalized sign of a permutation! $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 22, 2023 at 16:37
  • $\begingroup$ @IanGershonTeixeira That's exactly what it is, a sign of the permutation in some way. $n_{max}$ will be the lowest common multiple of the degrees of the reflections. So, for example, ST26 has a subgroup of index $6$. $\endgroup$
    – David A. Craven
    Commented Feb 22, 2023 at 16:38
  • $\begingroup$ Interesting example I stumbled on, ST25 of course has matrices of determinant $ 1, \zeta_3, \zeta_3^2 $ in its natural degree 3 reflection representation. But it also apparently has a different faithful degree 3 representation where all matrices have determinant 1. $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 22, 2023 at 20:41

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