2023/08/01 by Christof Geiß, Daniel Labardini-Fragoso, Geiß, Christof +3 · 1 citation
Mathematics · #13F60 #16G20 #57K20 #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2308.00792
openalex publication_date 2023/08/01 · openalex created_date 2023/08/18 · openalex updated_date 2026/07/28
Let \boldsymbolΣ:=(Σ,\mathbbM,ℙ) be a surface with marked points \mathbbM⊂∂Σ≠\varnothing on the boundary, and punctures ℙ⊂Σ∖∂Σ, and T an arbitrary tagged triangulation of \boldsymbolΣ 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) 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(\boldsymbolΣ)\rightarrowDecIrrτ(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(\boldsymbolΣ) is the set of laminations of \boldsymbolΣ 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-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)=\dimHomA(T)(-,τ(\bullet))+\dimHomA(T)(\bullet,τ(-)).