# Generalized root systems and reflection groups

Consider the following alternative definition of finite reflection group:

Definition: A finite reflection group $$\Gamma\subset\mathrm O(\Bbb R^d)$$ is a finite group generated by orthogonal transformations $$T\in\mathrm O(\Bbb R^d)$$ with eigenvalues $$\{-1^1,1^{d-1}\}$$. (the exponents denote multiplicites)

This definition suggests the following generalization:

Definition: A finite $$k$$-reflection group $$\Gamma\subset\mathrm O(\Bbb R^d)$$ is a finite group generated by orthogonal transformations $$T\in\mathrm O(\Bbb R^d)$$ with eigenvalues $$\{-1^k,1^{d-k}\}$$.

In other words: instead of inverting a 1-dimensional subspace, each generator inverts a $$k$$-dimensional subspace and leaves the orthogonal complement fixed.

As there are root systems associated with finite reflection groups, one can defined analogous systems for $$k$$-reflection groups. The elements of these are not vectors, but $$k$$-dimensional subspaces which are invariant w.r.t. certain "generalized reflections".

Question: Have such objects been studied before? Does there exist a classification?

Some thoughts

Let $$\Gamma$$ is a $$k$$-reflection group generated by "reflections" $$T_U,U\in\mathcal U$$, where $$\mathcal U$$ is the associated "generalized root system" that contains $$k$$-dimensional linear subspaces, and $$T_U$$ has eigenspace $$U$$ to eigenvalue $$-1$$. Then $$\Gamma'$$ generated by $$T_{U^\bot}=-T_U,U\in\mathcal U$$ is a $$(d-k)$$-reflection group.

So, all $$(d-1)$$-reflection groups are already classified via the usual reflection groups. In particular, up to dimension three, all generalized reflection groups are classified in this way. The first interesting case are the 2-reflection groups in $$\Bbb R^4$$, which are probably related to complex reflection groups.

• any complex reflection group in $GL_n$ gives an example with $k=2$ (just write it over $\mathbb{R}^{2n}$). – Dima Pasechnik Dec 2 '19 at 16:59
• @DimaPasechnik I understand that some complex reflection groups will give 2-reflection groups, but I think that the definition of complex reflection (that I take from here) is more general than what I have in mind here. E.g. they are not necessarily involutions. Am I right? But is the converse true? Every 2-reflection group comes from a complex reflection group? – M. Winter Dec 2 '19 at 17:06
• yes, you're right that there is no requirement for generators being involutions, I stand corrected. – Dima Pasechnik Dec 2 '19 at 17:10
• Obvious observations: these are quotients of Coxeter groups; but the finite Coxeter groups are exactly the finite real reflection groups; so, beyond that $k=1$ case, none of these will be full Coxeter groups (unless there is an accidental isomorphism with a $k=1$ case, I suppose). – Sam Hopkins Dec 2 '19 at 17:46
• If I understand correctly, if you place no restriction on $k$, a copy of any finite subgroup $G$ of $GL_d(R)$ generated by a conjugacy class of involutions will have to emerge from a classification, since $G$ is conjugate to a subgroup of $O_d(R)$. – Richard Lyons Dec 2 '19 at 18:26

If we place no restrictions on $$k$$, then this is precisely the class of finite groups that are generated by involutions.
In particular, if $$G$$ is any finite group of order $$n$$, then in the left regular representation of $$G$$ any involution acts as an $$n\times n$$ orthogonal matrix of order two and trace zero. Such a matrix must have eigenvalues $$1$$ and $$-1$$, each with multiplicity $$n/2$$, and is therefore a reflection across an $$(n/2)$$-dimensional subspace. Thus any group of order $$n$$ which is generated by involutions is an $$(n/2)$$-reflection group.
This class of groups is fairly large. For example, it includes all non-abelian finite simple groups. For if $$G$$ is a finite simple group, then the elements of order two in $$G$$ must generate a normal subgroup of $$G$$. If $$G$$ is non-abelian, then $$G$$ has even order and hence at least one element of order two, and therefore $$G$$ is generated by its elements of order two.