Reversals of autonomous subsets in right-angled Coxeter groups

Question

This question has to do with some experimental phenomenon in groups generated by involutions that I cannot explain.

Let $G$ be a finite, undirected graph, and let $W$ be the corresponding right-angled Coxeter group, i.e., the generators of $W$ are the vertices $v \in G$, and we have relations $v^2=1$ for all $v$ and $wv=vw$ if $v$ and $w$ are non-adjacent in $G$.

Recall that a Coxeter element $c$ in $W$ is a product of the generators $v$ in some order. It is well-known that Coxeter elements in such a right-angled Coxeter group correspond to acyclic orientations of $G$: for an acyclic orientation $\mathcal{O}$ of $G$, the corresponding Coxeter element $c$ is $c_{\mathcal{O}} = v_1v_2\cdots v_n$ where if $(v_i,v_j)$ is an arc of $\mathcal{O}$ then $i < j$. (This is enough to specify the Coxeter element because we can commute non-adjacent vertices.)

Furthermore, two Coxeter elements $c_{\mathcal{O}}$ and $c_{\mathcal{O}'}$ are conjugate in $W$ iff $\mathcal{O}'$ can be obtained from $\mathcal{O}$ by a series of sink-source reversals: choose a sink in $\mathcal{O}$ and reverse the direction of all arcs incident to that sink so that it becomes a source. In fact, there is even an explicit numerical criterion for deciding if $\mathcal{O}$ and $\mathcal{O}'$ are related by a series of sink-source reversals, having to do with choosing a basis of the space of cycles of $G$ (see Pretzel, "On reorienting graphs by pushing down maximal vertices", https://doi.org/10.1007/BF00390104).

My question is about a different kind of reversal, corresponding to something different than conjugacy in $W$.

Let $\mathcal{O}$ be an acyclic orientation of $G$; we say that a subset $A$ of vertices of $G$ is autonomous with respect to $\mathcal{O}$ if all vertices in $A$ have the same relationship to any vertex outside of $A$, i.e., for $a, a' \in A$ and $x \notin A$: we have an arc $(a,x)$ in $\mathcal{O}$ iff we have an arc $(a',x)$ in $\mathcal{O}$; we have an arc $(x,a)$ in $\mathcal{O}$ iff we have an arc $(x,a')$ in $\mathcal{O}$; and $a$ and $x$ are non-adjacent in $G$ iff $a'$ and $x$ are non-adjacent in $G$. For such an autonomous subset, let $\mathcal{O}_{-A}$ denote the orientation obtained from $\mathcal{O}$ by reversing all arcs inside of $A$ (i.e., between two vertices in $A$).

Question: Suppose that $W$ acts on a finite set $X$. Let $\mathcal{O}$ be an acyclic orientation of $G$ and let $A$ be an autonomous subset with respect to $\mathcal{O}$. Set $c := c_{\mathcal{O}}$ and $c' := c_{\mathcal{O}_{-A}}$. Is it true that the orbit structures of $c$ and $c'$ acting on $X$ are the same? In other words, is it true that, viewing the action as a homomorphism $\pi\colon W \to \mathfrak{S}_X$, we have that $\pi(c)$ and $\pi(c')$ are conjugate in $\mathfrak{S}_X$?

Probably it is inessential that $X$ be finite here, but for my purposes that would be enough.

Note that $c$ and $c'$ need not be conjugate in $W$: for example, with $A=G$, then $c'=c^{-1}$, and in general a Coxeter element will not be conjugate to its inverse. But certainly $c$ and $c^{-1}$ have the same orbit structure for any action.

EDIT: To make this a little more concrete, let me give one example of one example of an action of $W$ on a finite set.

Let $X$ be the set of independent sets of $G$. Recall that the generators of $W$ are the vertices $v \in G$. For $I\in X$ we define $$ v\cdot I = \begin{cases} I \cup \{v\} &\textrm{if $v\notin I$ and $I\cup \{v\}$ is an independent set}; \\ I\setminus \{v\} &\textrm{if $v\in I$}; \\ I &\textrm{otherwise}.\end{cases}$$ It's easy to check that this gives an action of $W$ on $X$, and in this case I can give an ad hoc argument that $c$ and $c'$ (related by a reversal of an autonomous subset) have the same orbit structure. But I suspect this is a more general phenomenon.

Answer 1

Okay, following the ideas from the comments, I do think there is a counterexample.

Let $G=K_4$, so $W$ is generated by involutions $v_1,\ldots,v_4$ subject to no other relations. Consider the Coxeter element $c=v_1v_2v_3v_4$. We can reverse the autonomous subset $\{v_1,v_2\}$ here to get $c'=v_2v_1v_3v_4$.

Now consider the homomorphism $\phi\colon W\to S_3$ defined by $v_1\mapsto (12)$, $v_2 \mapsto (13)$, $v_3\mapsto (13)$, $v_4\mapsto (23)$. Then $\phi(c)=(12)(13)(13)(23) = (231)$ while $\phi(c') = (13)(12)(13)(23) = e$, so obviously not conjugate.

Clearly what I was thinking could be the case was the wrong thing, but I still believe there should be a broader explanation for this phenomenon of autonomous subset reversal.