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4. Compactifications, tropical points, and $\Theta$-functions


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      Problem 4.1.

      Is there a natural way to compactify a braid variety so that the smaller strata also have cluster structures?


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          Problem 4.2.

          Can we obtain brick manifolds by cluster-theoretic means?


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              Problem 4.3.

              How does this relate to the combinatorics of the subword complex?
                  Comment: see Laura Escobar’s thesis


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                  Problem 4.4.

                  How does this relate to compactifications coming from tropical geometry?

                    •     For each cluster torus, take a co-character lattice and glue them by piecewise linear maps coming from exchange relations. These tropical points label the Gross-Hacking-Keel-Kontsevich $\Theta$-basis.

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                      Problem 4.5.

                      What are the tropical points of the braid variety? What are the $\Theta$-functions on the braid variety?

                        •     Let $U$ be a unipotent cell. Then $U$ can be thought of as a braid variety. Also $\mathbb{C}[U]$ has a Mirković-Vilonen-basis.

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                          Problem 4.6.

                          Does the MV-basis equal the $\Theta$-basis?
                              Comment: see the appendix of Baumann-Kamnitzer-Knutson.


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                              Problem 4.7.

                              What is the relationship between the piecewise linear combinatorics in the Casals-Gorsky-Gorsky-Le-Shen-Simental cluster algebra construction and the PL-combinatorics of MV-polytopes?

                                  Cite this as: AimPL: Cluster algebras and braid varieties, available at http://aimpl.org/clusterbraid.