Ordered Exchange Graphs

Understand

The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters.

• Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in non-positive degrees.
• These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support $\tau$-tilting modules and so on.
• All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects.