# Octonionic reflection groups

Consider Leech lattice definition provided by Wilson (Octonions and Leech lattice, 2008). There are 819 E8 sublattices defined by

$(2\lambda, 0, 0);$ $(\lambda \overline{s}, (\lambda \overline{s}) j, 0);$ $( (\lambda s)j, \lambda k, (\lambda j) k )$

where $\lambda$ span 240 vectors of E8 lattice, j,k are 16 base octonions (plus, minus), and s is -1+sum of imaginary unit octonions. (I am testing LaTeX here) See page 3, chapter 3 of Wilson paper. I wonder what is the subgroup of $Co_0$ generated by 819 reflections in 8-dim planes spanned by those E8 sublattices. They could be considered as octonion reflections. And as such they are elements of F4 Lie group being automorphism of $OP^2$.

My questions is following. Has anyone tried to extend definition of complex reflection and quaternion reflection to octonion reflection. In such definition Conway group $Co_0$ would be octonion reflection group i.e. it is generated by reflections in 8-dim planes in 24-dim Euclidean space.

In general when order 2 element in abstract group - called involution - can be considered as reflection ? I know involution is algebraic notion while reflection is geometric. But geometry is something which make group theory interesting.

Regards, Marek

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## 1 Answer

Yes, such groups are interesting, see this paper by Daniel Alcock, "Reflection groups on the octave hyperbolic plane," http://www.ma.utexas.edu/users/allcock/research/oh2.pdf

If you write to Daniel, he will probably give you more references.

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Thank you for the reference ! I have downloaded Allcock paper and also Serre lecture on finite subgroups of Lie groups. I will read them. I have made mistake thinking that 819 reflections in E8 spaces preserve 819 spaces. The first example is number 4 reflection (1,1,0) which does not preserve 70th E8 which is (lambda*s)i, lambda*j, (lambda*i)j. I checked this in GAP. This is nothing strange, because reflections 1,4,5,6,7,8,9,52 generate full Co0 which does not have presentation as 819 permutaions. Removing one by one generators we obtain Suzuki chain. Regards, Marek –  Marek Mitros Apr 12 '12 at 11:02
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