Laminations of Punctured Surfaces as τ-Regular Irreducible Components

Abstract Let $\mathbf{\Sigma }:=(\Sigma ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset \partial \Sigma \neq \emptyset $ on the boundary, and punctures $\mathbb{P}\subset \Sigma \setminus \partial \Sigma $, and $T$ an arbitrary tagged triangulation of $\mathbf{\Sigma }$ in the sense of Fomin–Shapiro–Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$ 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 $\pi _{T}\colon \operatorname{Lam}(\mathbf{\Sigma })\rightarrow \operatorname{DecIrr}^\tau (A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\operatorname{Lam}(\mathbf{\Sigma })$ is the set of laminations of $\mathbf{\Sigma }$, considered by Musiker–Schiffler–Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, $\operatorname{DecIrr}^\tau (A(T))$ denotes the set of generically $\tau $- 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 $E$-invariant of Derksen–Weyman–Zelevinsky, $E(-,\bullet )=\dim \operatorname{Hom}_{A(T)}(-,\tau (\bullet ))+\dim \operatorname{Hom}_{A(T)}(\bullet ,\tau (-))$.