Abstract
Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.
Highlights
Cluster algebras were introduced by Fomin and Zelevinsky in [14], and have since found applications across many types of mathematics
Cluster algebras were shown to be closely related to triangulations of surfaces by Fomin, Shapiro and Thurston in [13], where a quiver is constructed from a given triangulation and quiver mutations correspond to flipping an edge in the triangulation
A skew-symmetrizable matrix associated to a good orbifold with order 2 orbifold points can be unfolded to a skew-symmetric matrix associated to a surface which covers the orbifold. In this case we show that automorphisms of the marked exchange graph induce automorphisms of the unfolded exchange graph
Summary
Cluster algebras were introduced by Fomin and Zelevinsky in [14], and have since found applications across many types of mathematics. In the same paper Fomin and Zelevinsky defined the exchange graph of a cluster algebra to better visualise the combinatorics of the mutation class These graphs proved a useful tool in their classification of finite-type cluster algebras in [15] where these algebras were shown to correspond to Dynkin diagrams. The cluster automorphisms of any cluster algebra generated by mutationfinite skew-symmetrizable matrices can be studied using just the combinatorial properties of its marked exchange graph. We finish the paper with a conjecture generalising a result of Brustle and Qiu linking the tagged mapping class group of a surface with the cluster automorphisms of the corresponding surface cluster algebra.
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