论文标题
通用Caldero-chapoton函数具有系数和应用到表面群集代数的应用
Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras
论文作者
论文摘要
我们将Derksen-Weyman-Zelevinsky的表示突变视为表示空间上的密集定义的常规地图，并研究了具有系数的Caldero-Chapoton函数的通用值，例如，给出了足以使其线性独立性的足够组合条件。 For a quiver with potential $(Q,S)$, we show that if $k$ is a vertex not incident to any oriented 2-cycle, then every generically $τ$-reduced irreducible component $Z$ of any affine variety of (decorated) representations has a dense open subset $U$ on which Derksen-Weyman-Zelevinsky's mutation of representations $μ_k$ can be defined consistently as a regular map to an突变QP $μ_k（q，s）$的Jacobian代数的（装饰）表示的仿射品种。 我们的技术仅涉及基本的线性代数和基本代数几何形状，并且不需要假设雅各比·菲纳斯。因此，本文产生了通用Caldero-Chapoton功能突变不变性的新的，更通用的证据，对Derksen-Weyman-Zelevinsky和Plamondon的结果进行了推广并提供新的自然几何观点。 （对于具有潜在潜力的Jacobi-Finite非排成式砂纸，Plamondon使用Ginzburg DG-Algebras的机械和HOM-FINITE广义群集类别显示了这种不变性。） 我们将结果与磨坊，穆勒和秦的结果一起应用，以证明，对于任何选择的几何系数系统，不一定是完全排名，与可能刺破的表面相关的群集代数与边界上至少有两个标记点相关的是caldero-chapoton在边界上至少有两个标记的点，而caldero-chapoton可以用作laurent laurent polynomial coefferectients of lourent of coeffericatients的基础。
We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient combinatorial condition for their linear independence. For a quiver with potential $(Q,S)$, we show that if $k$ is a vertex not incident to any oriented 2-cycle, then every generically $τ$-reduced irreducible component $Z$ of any affine variety of (decorated) representations has a dense open subset $U$ on which Derksen-Weyman-Zelevinsky's mutation of representations $μ_k$ can be defined consistently as a regular map to an affine variety of (decorated) representations of the Jacobian algebra of the mutated QP $μ_k(Q,S)$. Our techniques involve only basic linear algebra and elementary algebraic geometry, and do not require to assume Jacobi-finiteness. Thus, the paper yields a new and more general proof of the mutation invariance of generic Caldero-Chapoton functions, generalizing and providing a new natural geometric perspective on results of Derksen-Weyman-Zelevinsky and Plamondon. (For Jacobi-finite non-degenerate quivers with potential, this invariance was shown by Plamondon using the machinery of Ginzburg dg-algebras and Hom-finite generalized cluster categories.) We apply our results, together with results of Mills, Muller and Qin, to prove that for any choice of geometric coefficient systems, not necessarily of full rank, the cluster algebra associated to a possibly punctured surface with at least two marked points on the boundary has the generic Caldero-Chapoton functions as a basis over the Laurent polynomial ring of coefficients.
