# Classification of octonionic reflection groups

I know that there exist classification theorems for real, complex, and quaternionic, reflection groups.

There are presentations for the real reflection groups, as well as further presentations for the complex ones. The ones for the quaternions and the complex numbers are less familiar, and, from what I can see, their Dynkin diagrams and Weyl symbols are not standardised like the real ones. There are also the three infinite families (discussed by Broué, Malle, and Rouquier in their paper on complex reflection groups).

I did try to see if anyone had attempted to classify the octonionic reflection groups. But as far as I could tell, no one has attempted to derive the complete set. If this has already been done, then please refer me to an open access source. If not then this’ll probably be an open problem, but I am nowhere near experienced enough in the topic of reflection groups to even attempt a classification.

Here is the paper on complex reflection groups: https://www.math.ucla.edu/~rouquier/papers/banff.pdf

• As far as I can tell, the two lists don't add to the question and just make it longer than necessary, so you should remove them. It would be more helpful to include a definition of octonionic reflection group. Sep 5, 2021 at 22:22
• I don’t know the exact definition of an “octonionic reflection”. As I said, I am rather inexperienced. All I could find were “octonionic hyperbolic reflection groups” which are not what I am asking about. I am asking about finite groups. Sep 5, 2021 at 22:30
• Are octonionic reflection groups supposed to be associative (like groups) or not associative (like the octonions)? // Could you post your reference on octonionic hyperbolic reflection groups? Sep 5, 2021 at 22:53
• I did say octonionic reflection groups which would by definition, imply associativity. here is the reference to hyperbolic octonionic reflection groups: ma.utexas.edu/users/allcock/research/oh2.pdf Sep 6, 2021 at 0:30
• It seems a hyperbolic octonionic reflection groups are contained in the symmetry group of the hyperbolic octonionic plane, so finite octonionic reflection groups are contained in the symmetry group of the (compact) hyperbolic octonionic projective plane, which is the compact form of $F_4$. Specifically, I think a reflection is supposed conjugate to an order $2$ element of $\mathrm{Spin}_9 \subset F_4$ that's not central. So this gives a definition: a finite subgroup of $F_4$ generated by elements in that conjugacy class. Sep 7, 2021 at 1:15