I asked a version of this question in Math StackExchange about a week ago but I've received no feedback so far, so following the advice I received on meta I decided to post it here.


The irreducible finite complex reflection groups were classified by Shephard and Todd. The list consists of a three-parameter family of imprimitive groups and 34 exceptional cases. If $\mathbb{K}$ is any field of characteristic zero, it is known (see e.g. Section 15-2 here) that if a group has a representation as a $\mathbb{K}$-reflection group then it also has a representation as a complex reflection group, so we get no new examples for fields. I'm interested to know if an analogue of this classification exists when we allow $\mathbb{K}$ to be a division ring of characteristic zero.

I have only found partial results. The irreducible finite quaternionic reflection groups were classified in this paper by Cohen. Finite $\mathbb{K}$-reflection groups of rank $1$, or equivalently finite subgroups of division rings of characteristic $0$, were classified in this paper by Amitsur. The classification for rank $2$ can perhaps be extracted from the classification of finite subgroups of $GL(2,\mathbb{K})$ by Banieqbal using a case-by-case check.

In view of the complex and quaternionic lists, I would expect the full classification (if there is one) to follow a form similar to this:

  • An infinite family $G_n(M,P,\alpha)$ of imprimitive reflection groups of rank $n$, where $M$ is a finite subgroup of a division ring $\mathbb{K}$ (i.e. an Amitsur group) and $[M,M]\le P \trianglelefteq M$, possibly with some extra data $\alpha$ in low rank. Algebraically it should correspond to something like the group of generalized permutation matrices with entries in $M$ whose determinant is in $P$, like in the cases of fields and quaternions (note that order doesn't matter when computing the determinant, since $[M,M]\le P$).

  • A family or families of examples in rank $2$ (or perhaps in rank $\le m$ where $m^2$ is the dimension of $\mathbb{K}$ as a division algebra over its center). In the quaternionic case they are the primitive reflection groups whose complexification is imprimitive, and are all constructed from certain $2$-dimensional primitive complex reflection groups; it's not clear to me how this construction could generalize to arbitrary $\mathbb{K}$.

  • A number of exceptional cases of small rank. These are the ones I'm most interested in. In the quaternionic case these are precisely the primitive reflection groups whose complexification is also primitive; in the general case they might correspond to primitive reflection groups which remain primitive after tensoring the representation with a splitting field.


My main question is thus:

Is there a classification of groups representable as a $\mathbb{K}$-reflection group over some division ring $\mathbb{K}$ of characteristic zero?

I would also appreciate any references dealing with this problem or with particular cases. If the classification turns out to be intractable, or currently out of reach, I would ask if at least an example can be found of a new exceptional reflection group of rank $\ge 3$ (see details above).


After much searching, I finally found a brief reference to this problem in the literature. The mention occurs at the end of Section 3 in this 1981 paper by Kantor, which I quote here for convenience:

[...] For example, consider the problem of determining all finite primitive reflection groups $G$ in $GL(n,D)$, for $D$ an arbitrary noncommutative division ring of characteristic $0$. If $n=1$, this is just the famous problem solved by Amitsur (1955) (and independently and almost simultaneously by J. A. Green). If $n=2$ and $G$ is solvable, the problem seems to involve even more difficult number theory than Amitsur used. But if $n\ge 3$, and if simple group classification theorems are thrown at the problem, no new nonsolvable examples arise. [...]

If the last sentence is true, it means that the new examples of "exceptional groups" I asked for must necessarily be solvable. However, the author does not provide any in-text citation for that statement, and I haven't been able to find which result is being alluded to.


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