Laminations of punctured surfaces as $\tau$-regular irreducible components

Abstract

Let $\mathbf{\Sigma }:=(\Sigma ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset \partial \Sigma \ne \varnothing$ 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 $W(T)$ 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 ${\pi }_T:\mathrm{Lam}(\mathbf{\Sigma })\to {\mathrm{DecIrr}}^{\tau }(A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\mathrm{Lam}(\mathbf{\Sigma })$ 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, ${\mathrm{DecIrr}}^{\tau }(A(T))$ 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, $E(-,\bullet )=\dim {\mathrm{Hom}}_{A(T)}(-,\tau (\bullet ))+\dim {\mathrm{Hom}}_{A(T)}(\bullet ,\tau (-))$.

arXiv commentv2: Main result vastly generalized, from tagged triangulations of signature zero, to arbitrary tagged triangulations; v3: Changed terminology from "$\tau$-reduced" to "$\tau$-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

References in corpus (14)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Quivers with potentials associated to triangulated surfaces
• Bases for cluster algebras from surfaces
• The representation type of Jacobian algebras
• Cluster categories for marked surfaces: punctured case
• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• Generic bases for cluster algebras and the Chamber Ansatz
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• On the combinatorics of gentle algebras
• The derived category of surface algebras: the case of the torus with one boundary component
• The Goldman bracket determines intersection numbers for surfaces and orbifolds
• Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras
• Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary
• Generically $\tau$-regular irreducible components of module varieties

Cited by in corpus (1)

• On homomorphisms and generically $\tau$-regular components for skewed-gentle algebras