Defining a Coxeter group using all reflections

Question

Let $(W, S)$ be a pair of a group $W$ and a subset $S$ consisting of involutions of $W$. We can consider the group $\tilde W$ with presentation $\langle S | \mathcal{R} \rangle$ where $\mathcal{R}$ consists of the relations $(st)^{\operatorname{ord}(st)}$ for $s, t \in S$ (the order computed in $W$, of course). If the group homomorphism that sends each $s \in \tilde W$ to the corresponding $s \in W$ is an isomorphism, $(W, S)$ is called a Coxeter system.

The elements $s \in S$ are called simple or fundamental reflections but every element of the form $wsw^{-1}$ ($s \in S, w \in W$) deserves the name reflection as well. Let $R$ be the system of all reflections of $(W, S)$.

Is there a nice characterization of pairs $(W, R)$ that arise this way? By “nice” I mean that I don’t want to choose simple reflections first.

My main aim is to better understand Weyl groups and root systems/data and their relation to parabolic subgroups of (algebraic) reductive groups, but again without necessarily choosing positive roots or a minimal parabolic subgroup. In particular, I would be content with an answer that only applies to finite groups as well.

Answer 1

What you're asking about is what's sometimes called the "dual approach" to reflection groups and their braid groups. A good place to start would be

• D. Bessis, The Dual Braid Monoid, https://doi.org/10.1016/j.ansens.2003.01.001 and https://arxiv.org/abs/math/0101158

and go reference-chasing from there (including the Birman--Ko--Lee and Brady and Brady--Watt papers Bessis cites). In particular, Bessis's Question 1.1.1 (in the arXiv version, I'm not sure if the numbering is the same in the published version) is morally equivalent to your question.

Answer 2

There are probably different angles to look at what constitutes a "reflection". My favorite is a transformation in $n$-dimensional space that preserves an $n-1$-dimensional subspace. The "root" can be associated with the 1-dimensional subspace that isn't preserved. You can add restrictions that its order=2 which gives the normal definition. Some generalization allow other orders which gives complex reflection groups...You can probably generalize when $n-2$-dimensional subspace is preserved...though the $n-1$ case is already very rich in structure. So all reflections are defined and you can take them to be the generators of the group; the "simple" roots just give you a minimal set of generators.

Here's an example for the "order matrix" using simple reflections only and all of them. (per comments below)

orders_simple:=[
[1,3,2],
[3,1,3],
[2,3,1]];

orders_full:=[
[1,3,2,3,3,3],
[3,1,3,3,3,2],
[2,3,1,3,3,3],
[3,3,3,1,2,3],
[3,3,3,2,1,3],
[3,2,3,3,3,1]];