The combinatorics of cluster algebras is encoded in the module categories of 2-Calabi-Yau (2-CY) tilted algebras, which then satisfy many nice properties. In particular, their syzygy modules form a 3-CY triangulated category, which in this setting is equivalent to the category of Cohen-Macauley modules and also the singularity category of the algebra. We find a geometric model for this category for a certain class of 2-CY tilted algebras defined by quivers with relations. More precisely, we construct a decorated polygon with a checkerboard pattern whose 2-diagonals correspond to syzygies. Moreover, other representation theoretic aspects such as morphisms, extensions, Auslander-Reiten triangles, and the shift also have a geometric interpretation in this polygon. This is joint work with Ralf Schiffler.
Speaker(s): Khrystyna Serhiyenko (University of Kentucky)