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Five years ago, Gross-Hacking-Keel-Kontsevich made a major advance in the theory of cluster algebras, by constructing bases of cluster algebras in large generality. A key tool in their construction is the theory of scattering diagrams.

Some examples of scattering diagrams are described in the literature, especially for rank 2 cluster algebras. However, for me they remain somewhat mysterious.

I am interested in cluster algebras arising from semisimple groups, especially the algebra $ \mathbb C[N] $ of functions on the maximal unipotent subgroup of a semisimple group? (For example, for $ SL_n$, this is the group of uni-upper triangular matrices.)

Question:

Has anyone has described/computed the scattering diagram associated to the cluster algebra $ \mathbb C[N] $?

If this example is too complicated, I would also be happy with understanding the scattering diagrams associated to the subalgebras $ \mathbb C[N(w)] $ or the coordinate rings of partial flag varieties.

Updated question:

Since $ \mathbb C[N] $ is not a finite-type cluster algebra (unless $ G = SL_2, \dots, SL_5$) it seems to be impossible to completely describe the scattering diagram. However, perhaps it is possible to give a rough description of it, for example in Figure 1.3 of GHKK, they describe the appearance of the rank 2 scattering diagrams of non-finite type. Is it possible to give an analogous description for $ \mathbb C[N] $?

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  • $\begingroup$ Does arxiv.org/abs/1502.03769 help? $\endgroup$
    – David E Speyer
    Commented Jun 23, 2019 at 18:00
  • $\begingroup$ I've read that article, which partially piqued my interest in the subject. However, it doesn't compute the scattering diagram. (Also, there is an error in the description of the theta basis in that paper.) $\endgroup$
    – Joel Kamnitzer
    Commented Jun 23, 2019 at 19:56
  • $\begingroup$ I should add that Magee's article (which David linked to) is quite nice, as his followup one (front.math.ucdavis.edu/1709.05776) which treats invariants in triple tensor products. These papers show that the GHKK theory applies to the representation theoretic situations. However, they don't give any description of the resulting bases or the scattering diagrams. $\endgroup$
    – Joel Kamnitzer
    Commented Jun 24, 2019 at 6:53
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    $\begingroup$ Since I can't be sure the answer to the question as asked is no, I'm putting this in a comment instead: I don't believe so. For $n>5$ (or possibly $6$; I'm always out by one), the cluster structure is "wild" - infinite type, and very complicated, just like the canonical basis is. For very small $n$, the answer will be yes (see e.g. a talk of Mark Gross, dpmms.cam.ac.uk/~mg475/clusters.pdf) because they're finite type and easy. $\endgroup$
    – Jan Grabowski
    Commented Jun 25, 2019 at 10:40
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    $\begingroup$ Also, in my view, I don't think we really know very much explicit about the $\mathbb{C}[N]$ cluster algebras beyond suitable initial data, as Geiss-Leclerc-Schroer's approach is more sophisticated and geometric. I don't think I could easily "read off" information you'd want for a scattering diagram. (But perhaps others who know better can.) $\endgroup$
    – Jan Grabowski
    Commented Jun 25, 2019 at 10:41

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