Quiver folding and maximal green sequences The technique of quiver folding (please see Folding by Automorphisms) can be used to prove statements about non-simply laced quivers (i.e. valued quivers) when they are already known in the simply-laced case. 
I wonder whether we can use this technique to fold maximal green sequences of a simply-laced quiver $Q$ (for example $A_3$ alternating orientation) to maximal green sequences in the respective valued quiver $Q'$ that can be produced by folding $Q$ (for example $B_2=C_2$). May I ask whether this can be done? Thank you very much!
 A: This will certainly work fine in finite type.  Folding $Q$ to $Q'$ corresponds to an inclusion of $W'$ into $W$, where the reflections of $W'$ are mapped to products of commuting reflections in $W$.  $c'$-sortable elements of $W'$ give you a sublattice of the $c$-sortable elements of $W$.  Thus, if you take a maximal green sequence for $W'$, you get an increasing sequence of $c$-sortable elements in $W$.  You can then refine this to a maximal green sequence in $W$.  You can typically refine it in more than one way, of course.
(Edited to add:) Conversely, suppose we start with a maximal green sequence for $W$.  There is a lattice quotient of the $c$-Cambrian lattice which sends each $c$-sortable to the maximum $c'$-sortable in the image of $W'$ below it.  Since this is a lattice quotient, general results of Reading imply that it corresponds to a coarsening of the $c$-Cambrian fan for $W$, which is isomorphic to the $c'$-Cambrian fan.  The maximal green sequence for $W$ can be thought of as a positively oriented path through the $c$-Cambrian fan.  It induces a positively oriented path through the coarsened fan, which is a maximal green sequence for $W'$.  
This gives us a way to fold any maximal green sequence for $W$ to obtain a maximal green sequence for $W'$.
