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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.