Laminations of punctured surfaces as τ-reduced irreducible components

Abstract

Let Σ:=(Σ,M,P) be a marked surface with marked points on the boundary M⊂∂Σ≠∅, and punctures P⊂Σ∖∂Σ, and let T be signature zero tagged triangulation of Σ in the sense of Fomin-Shapiro-Thurston. In this situation the corresponding non-degenerate Jacobian algebra A(T):=PC(Q(T),W(T)) is skewed-gentle. Building on ideas by Qiu-Zhou, and on recent progress concerning the description of homomorphisms between representations of skewed-gentle algebras by the first author, we show that there is an isomorphism πT:Lam(Σ)→DecIrrτ(A(T)) of tame partial KRS-monoids, which intertwines generic g-vectors and shear coordinates with respect to T. Here, Lam(Σ) is the set of laminations of Σ considered by Musiker-Schiffler-Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, DecIrrτ(A(T)) denotes the set of generically τ-reduced irreducible components of the decorated representation varieties of A(T), with the direct sum of generically E-orthogonal irreducible components as partial monoid operation, where E is the symmetrized E-invariant of Derksen-Weyman-Zelevinsky.