Combinatorics

The discovery of cluster algebras, and in particular the Cartan-Killing type classification of cluster algebras of finite type [57] led to natural generalizations of classical objects in combinatorics. For example the Catalan numbers count the number of clusters for a cluster algebra of type $\mathsf{A}$, while the Stasheff associahedra and their polytopal realization describe in this case the dual of the cluster complex (and the Bott-Taubes cyclohedra do the same for type $\mathsf{B}$). These concepts could now be associated to any root system of finite type [58] and [40], see also [52] for an introduction. This triggered an intensive research in this direction, see for example the papers by:

Athanasiadis et al.: [8,7,6,10,9,11]; Bell and Skandera: [13]; Bergeron et al.: [16]; Bessis et al.: [17,18]; Carr et al.: [34]; Carrol and Speyer: [35]; Chapoton et al.: [36,39,38,37], Drake, Gerrish and Skandera: [46,45]; Fomin and Reading: [53]; Krattenthaler: [81]; McCammond: [88]; Musiker and Propp: [89]; Panyushev: [91]; Reading: [96,95,94]; Reiner, Welker: [97]; Sandman: [100]; Simion: [102]; Speyer: [104]; H. Thomas: [107]; Tzanaki: [109]; Zelevinsky: [112];

Also, inspired by cluster algebras is the work of L. Williams et al. around the combinatorics of tropical geometry and total positivity [103,111,93,82].

In a different direction, the search for efficient (combinatorial) criteria to recognize cluster algebras of finite type was started by A. Seven's (2004) thesis [101], and motivated some interesting combinatorics with connections to representation theory of algebras, see for example [12], [30], and [29].

Finally, one should mention that many problems for cluster algebras can be stated in purely combinatorial terms, see [59] for a remarkable collection. However, their solution might require deep results from geometry and representation theory. Examples are the papers by Caldero et al.:[33,32], where the coefficients of the Laurent expansion of cluster variables are interpreted as Euler characteristics of quiver Grassmanians, and Lusztig's canonical basis is used to prove their positivity. Other examples are the recent papers by Keller et al. [43,60] where special cases of several of the conjectures in loc. cit. are studied via the structure of certain 2-Calabi-Yau categories.