Reflection quotients of Coxeter groups

Question

I am interested in a concept somehow dual to reflection subgroups. A reflection quotient of a Coxeter system $(W, S)$ shall be a surjective homomorphism $W \to W'$ to a Coxeter group $W'$ such that the images of $S$ are reflections. I did not find any literature on the topic, so it's possible that this notion has a name different from the one I just gave it. My question basically is the following:

Given a Coxeter system $(W, S)$, is there a way to find all its reflection quotients?

For $S_n$ (the Coxeter group with graph $A_{n-1}$), the answer is quite straightforward because $S_n$ has almost no normal subgroups: The only reflection quotients are $S_n$ and $S_2$. I do not know an answer for any other Coxeter groups, not even the finite ones. I am particularly interested in the Coxeter groups with graph $D_n$ (I even asked a specific question on it on MSE, but it did not receive any attention so far). The general problem however seems interesting enough for this forum, I hope.

One would hope that the problem has a graph-theoretic answer, similar to the one requested in the dual question to this one. I do not know how realistic this hope is.

Answer 1

For right-angled Coxeter groups, the "easy case", an answer can be deduced from my article Morphisms between RACGs and the embedding problem in dimension two. Theorem 4.2 states that:

Theorem. Let $\Gamma_1,\Gamma_2$ be two finite graphs and $\rho : C(\Gamma_1) \to C(\Gamma_2)$ a morphism. Assume that $\rho(u)$ is a reflection for every $u \in V(\Gamma_1)$. There exist

• a sequence of graphs $\Lambda_0=\Gamma_1, \Lambda_1, \ldots, \Lambda_k$;
• partial conjugations $\alpha_1 \in \mathrm{Aut}(C(\Lambda_1)), \ldots, \alpha_k \in \mathrm{Aut}(C(\Lambda_k))$;
• foldings $\pi_1 : C(\Lambda_0) \to C(\Lambda_1), \ldots, \pi_k : C(\Lambda_{k-1})\to C(\Lambda_k)$;
• and a peripheral embedding $\bar{\rho} : C(\Lambda_k) \hookrightarrow C(\Lambda_k)$

such that $\rho = \bar{\rho} \circ \alpha_{k-1} \circ \pi_{k-1} \circ \cdots \circ \alpha_1 \circ \pi_1$.

A morphism $C(\Phi) \to C(\Psi)$ is a folding if there exists an equivalence class $\sim$ on the vertices of $\Phi$ such that $\Psi = \Phi / \sim$ and such that $\rho$ sends each generator of $\Phi$ to the generator of $\Psi$ corresponding to the equivalence class of the associated vertex of $\Phi$. For your question, the key consequence of this theorem is that:

Corollary. Let $\Phi,\Psi$ be two finite graphs. There exists a reflection quotient map $C(\Phi) \twoheadrightarrow C(\Psi)$ if and only if $\Psi$ can be obtained from $\Phi$ by collapsing vertices.

Answer 2

It turns out that for finite Coxeter groups $W$, this was solved by Maxwell in 1996 in the article The normal subgroups of finite and affine Coxeter groups. His Theorem 0.1 asserts that

Theorem. If $W$ is an irreducible finite Coxeter group and $H$ is a normal subgroup of $W$, then either $H = \{ \pm 1 \}$, or there exists a graph homomorphism $\psi \colon W \to W'$ such that $H = \ker(\psi)$.

Here, a graph homomorphism maps standard generators of $W$ either to $1$ or to a standard generator of $W'$ (so a graph homomorphism is a reflection quotient map if and only if none of the generators map to $1$). Together with the classification of finite Coxeter groups, this makes it rather straightforward to list the reflection quotients (in fact, this is done in the same article in Table 3).

The answer in the case of affine Coxeter groups seems more complicated.