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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!

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  • $\begingroup$ Have you tried some examples? Keller's Java applet (available on his home page) has maximal green sequence functionality. $\endgroup$
    – Jan Grabowski
    Jun 20, 2016 at 10:01
  • $\begingroup$ @JanGrabowski I tried to fold $A_3$ into $B_2$ and look at the maximal green sequences. It does not work very well. $\endgroup$
    – Ying Zhou
    Jun 20, 2016 at 16:37
  • $\begingroup$ You mean the folding doesn't seem to work? This wouldn't be too surprising to me. (But that's only instinct, I don't have any formal obstructions to raise immediately.) $\endgroup$
    – Jan Grabowski
    Jun 22, 2016 at 11:58

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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 $W'$.

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  • $\begingroup$ I was thinking about the problem several days before your answer appeared using the maximal green sequences of $A_3$ alternating orientation and semi-invariant pictures. The interesting part is that not all mutation sequences in general and maximal green sequences in particular can be folded. Only those that are in some sense symmetric can be folded. $\endgroup$
    – Ying Zhou
    Jul 5, 2016 at 0:26

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