4. Compactifications, tropical points, and \Theta-functions
-
Problem 4.1.
Is there a natural way to compactify a braid variety so that the smaller strata also have cluster structures? -
Problem 4.3.
How does this relate to the combinatorics of the subword complex? -
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.
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.
Problem 4.6.
Does the MV-basis equal the \Theta-basis? -
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.