Cluster structures in geometry: Construction, investigation, applications to cluster combinatorics
Final Report Abstract
The project consists of three main directions (1) structure theory of cluster algebras of finite mutation type, (2) connections of cluster algebras to Coxeter groups and (3) cluster-type structure on double pants decompositions of surfaces). All of these directions enjoy a good progress. The main results of the work are the following. 1. Cluster algebras of finite mutation type. Together with Shapiro and Tumarkin, we build a geometric realization of almost all cluster algebras of finite mutation type. Namely, all but finitely many cluster algebras of finite mutation type are cluster algebras arising from a triangulated orbifold (manifold in skew-symmetric case). This realization is useful to extend numerous theorems known for the cluster algebras arising from surfaces to more general orbifold setting. The orbifold realization of almost all cluster algebras of finite mutation type enables us to find the growth rates of almost all cluster algebras of finite mutation type. Together with Shapiro, Thomas and Tumarkin, we find the growth rates of all cluster algebras. 2. Cluster algebras and Coxeter groups. Writing the project, we expected cluster algebra (with chosen initial seed) correspond to a certain Coxeter group (with a certain choice of generating reflections), while the mutation corresponds to the conjugation of some generators. We expected that this property is general for a large class of cluster algebras. Recently, Barot and Marsh proved this property for the case of cluster algebras of finite type. As a generalization we (together with Tumarkin) extended the statement to the case of affine cluster algebras. Furthermore, for the case of cluster algebra arising from a surface or an orbifold without punctures, we realized that the similar statement holds (the only difference is that now one has some quotient of a Coxeter group). This part of work is still in progress. The established connections induced a deeper study of reflection subgroups of Coxeter groups. More precisely, together with Fintzen and Tumarkin, we found an explicit criteria when an odd-angled Coxeter group has a finite index reflection subgroup (the Coxeter diagrams of groups with finite index reflection subgroups satisfy some easy to check properties). 3. Cluster-like structure on double pants decompositions of surfaces. Together with Natanzon we introduced a cluster-like structure on the double pants decompositions of surfaces corresponding to the Heegaard splittings of the 3-sphere. In particular, the counterparts of the cluster variables are the lengths of closed geodesics realizing the pants decomposition for hyperbolic surfaces. The lengths u of the curves lying in the same double part decomposition define the surface (as a point of the Teichm¨ller space) up to finitely many choices. The local charts arising from different double pants decompositions compose an atlas covering the Teichmüller space. The gluings of the adjacent charts are coming from the elementary transformations of the decompositions, the gluing functions are algebraic. The same charts provide an atlas for a large part of the boundary strata in Deligne-Mumford compactification of the moduli space.
Publications
- Cluster algebras and triangulated orbifolds, Advances in Mathematics 231 (2012), 2953–3002
A. Felikson, M. Shapiro, P. Tumarikin
- Growth of cluster algebras
A. Felikson, M. Shapiro, H. Thomas, P. Tumarkin
- Moduli via double pants decompositions, Differential Geometry and its Applications 30 (2012), 490-508
A. Felikson, S. Natanzon
- Reflection subgroups of odd-angled Coxeter groups
A. Felikson, J. Fintzen, P. Tumarkin
