This is a place to explore and summarize the important information in a cluster algebra. It is currently maintained by Zack Greenberg
Currently you explore the codimension 1 subalgebras, isomorphism classes of quivers and in some cases orbits of X-coordinates. Each page shows the full cluster complex for the subalgebra. You can then "focus" on various features by choosing the appropriate button.
We group the subalgebras by their "cluster type" following the Lie Algebra type classification of finite cluster algebras by Fomin and Zelevinsky. In addition each codimension 1 subalgebra is labeled by the single A coordinate that was frozen to create the subalgebra.
In cluster algebras associated to Gr(2,n) all the A coordinates are "Plucker coordinates." For example "p34" corresponds to the function that take the determinant of the third and fourth columns of the 2xn matrix associated to a point in Gr(2,n)
The X coordinates in Gr(2,n) correspond to "cross ratios". They depend on 4 indices and so we write x1234 as short hand for (p12 * p23)/(p14 * p23). The seeds of Gr(2,n) are in bijective correspondence with triangulations of an n-gon. Under this correspondence pij corresponds to the diagonal from vertex i to vertex j. Each X coordinate corresponds to a quadrilateral. The cluster modular group corresponds to rotation of the regular n-gon.
In Gr(3,6) every A coordinate is either a Plucker coordinate pijk or one of two exotic coordinate e2x or e2y. See Scott's paper for more information on these coordinates.
In Gr(3,7) every A coordinate is either a Plucker coordinate pijk or one a version e2x or e2y corresponding to 6 of the 7 possible columns. For example e2x3 corresponds to e2x if we removed the third column of the 2x7 matrix. See Scott's paper for more information on these coordinates.
All of these cluster algebras are infinite and so we can only show a finite subset of the cluster complex. The infiniteness is due to the action of the mapping class group of the annulus, which is generated by a Dehn twist around the center. As such the cluster complex repeats the pattern shown in both directions.
The A coordinates correspond to arcs between the p marked points on the inner boundary component and the q marked points on the outer boundary. We label the marked points 0 to p-1 and 0 to q-1 clockwise around each boundary. An arc with both endpoints on the inner boundary component is labeled iXYw0 for the clockwise arc from X to Y and iXYw1 for the counterclockwise arc. Similarly oXYw0 is a clockwise arc between two outer point and oXYw1 is counterclockwise. Edges cXYwN are edges from inner point X to outer point Y winding N times around the annulus.