Laminations of Punctured Surfaces as tau-Regular Irreducible Components

Let ∑ := (∑, M, P) be a surface with marked points M ⊂ ∂∑ ≠= ∅ on the boundary, and punctures P ⊂ ∑ \ ∂∑, and T an arbitrary tagged triangulation of ∑ in the sense of Fomin–Shapiro–Thurston. The Jacobian algebra A(T) := P(Q(T), W(T)) corresponding to the non-degenerate potential W(T) of T defined by Cerulli Irelli and the second author is tame, as shown by Schröer and the first two authors. In this paper, we show that there is a natural isomorphism πT : Lam(∑) → DecIrrτ (A(T)) of tame partial KRS-monoids that intertwines dual shear coordinates with respect to T, and generic g-vectors of irreducible components. Here, Lam(∑) is the set of laminations of ∑, considered by Musiker–Schiff ler–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 - regular 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 Einvariant of Derksen–Weyman–Zelevinsky, E(−, •) = dim HomA(T)(−, τ (•)) + dim HomA(T)(•, τ (−)).