论文标题
等级的建设$ 2 $ grassmannian集群类别中的不可塑性模块
Construction of Rank $2$ Indecomposable Modules in Grassmannian Cluster Categories
论文作者
论文摘要
The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \ cite {JKS16}。在此类别中的不可分解的模块中，有$ 1 $的模块，该模块与$ \ {1,2，\ dots，n \} $一起进行$ k $ -subsets，其显式结构由Jensen，King和su提供。这些是类别的构建块，因为$ {\ rm cm}中的任何模块（b_ {k，n}）$可以被它们过滤。在本文中，我们给出了等级2模块的明确构造。这样，在$ k = 3 $和$ k = 4 $的情况下，我们给出了所有难以解决的等级2模块。特别是，我们涵盖了驯服的案件并超越了它们。我们还表征了它们之间的模块，这些模块是由它们的过滤决定的。对于$ k \ ge 4 $，我们展示了具有相同过滤的非晶状体等级2模块的无限家族。
The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by them. In this paper we give an explicit construction of rank 2 modules. With this, we give all indecomposable rank 2 modules in the cases when $k=3$ and $k=4$. In particular, we cover the tame cases and go beyond them. We also characterise the modules among them which are uniquely determined by their filtrations. For $k\ge 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.
