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3. The peripheral locus

Let X be a cluster variety. Let T_1,T_2,\dots be the cluster tori. Let P denote the complement X-\cup T_i.


    1. Add remark

      Problem 3.1.

      When is P empty (in G(2, 2k+1))?

      What about the big open positroid cell in G(k,n)?
        1. Remark. [José Simental] P is always empty for the big positroid cell in Gr(2,2k+1).


            • Add remark

              Problem 3.2.

              When P is nonempty, when do we expect the complement of P to be codimension 2?

                •     Let X be really full rank. This implies there is a torus of dimension the number of frozen variables acting on X preserving cluster structure. S acts freely on each cluster torus so any point with nontrivial S-stabilizers is in P.

                  In G(k,n), nontrivial stabilizers correspond to disconnected matroids. So if [v_1,\dots, v_n] is a k\times n matrix where the matroid of v_1,\dots, v_n is disconnected, this point of G(k,n) is in P.

                  Add remark

                  Problem 3.3.

                  Is the converse also true (Is P for the big positroid cell exactly the points with disconnected matroid)?

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