Laminations of punctured surfaces as $\ddot{I}„$-regular irreducible components

Abstract

Let $\hat{\mathbf{I}}\pounds :=(\hat{I}\pounds ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset \partial \hat{I}\pounds \ne \varnothing$ on the boundary, and punctures $\mathbb{P}\subset \hat{I}\pounds \setminus \partial \hat{I}\pounds$, and $T$ an arbitrary tagged triangulation of $\hat{\mathbf{I}}\pounds$ in the sense of Fomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T),W(T))$ corresponding to the non-degenerate potential $W(T)$ 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 $\ddot{I}{€}_T:\mathrm{Lam}(\hat{\mathbf{I}}\pounds )\to {\mathrm{DecIrr}}^{\ddot{I}}„(A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\mathrm{Lam}(\hat{\mathbf{I}}\pounds )$ is the set of laminations of $\hat{\mathbf{I}}\pounds$ considered by Musiker-Schiffler-Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, ${\mathrm{DecIrr}}^{\ddot{I}}„(A(T))$ denotes the set of generically $\ddot{I}„$-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, $E(-,\bullet )=\dim {\mathrm{Hom}}_{A(T)}(-,\ddot{I}„(\bullet ))+\dim {\mathrm{Hom}}_{A(T)}(\bullet ,\ddot{I}„(-))$.

v2: Main result vastly generalized, from tagged triangulations of signature zero, to arbitrary tagged triangulations; v3: Changed terminology from "$\ddot{I}„$-reduced" to "$\ddot{I}„$-regular", including the title. After referee report many corrections, more examples, added indices for notation and symbols. 46 pages, 13 figures. To appear in IMRN