论文标题
刻有格拉曼尼亚集群类别中的不可分解的模块
Rigid Indecomposable Modules in Grassmannian Cluster Categories
论文作者
论文摘要
$ \ Mathbb {C}^n $中的Grassmannian品种的坐标环具有群集代数结构，并带有Plücker关系，从而引起了交换关系。在本文中，我们研究了相应的Grassmannian集群类别的不可分解模块$ {\ rm cm}（b_ {k，n}）$。 Jensen，King和SU已将Kac-Moody root System $ J_ {K，N} $与$ {\ rm cm}（b_ {k，n}）$相关联，并在有限类型中表明，刚性类型，刚性不可兼容的模块对应于根。通常，类别$ {\ rm cm}（b_ {k，n}）$与root System $ j_ {k，n} $之间的链接仍然是神秘的，这是一个开放的问题，这是否始终赋予根源。在本文中，我们在无限类型中提供了这种关联的证据：我们表明，每个不可分解的等级2模块都对应于关联的根系的根。我们还表明，$ {\ rm cm}中的不可分配等级3个模块（b_ {3，n}）$都产生了$ j_ {3，n} $的根。对于$ {\ rm cm}中的等级3模块（b_ {3，n}）$对应于真正的根，我们表明它们的基础配置文件是某个规范的循环排列。我们还以$ {\ rm cm}（b_ {3，n}）$对应于假想根的级别3模块。通过证明恰好有225个刚性不可兼容的等级3个模块，其中包括$ {\ rm cm}（b_ {3,9}）$，我们在这种情况下确认了Grassmannian群集类别和关联的根系之间的链接。我们推测，与真实根相对应的$ {\ rm cm} $ {\ rm cm} $中的任何刚性不可传输模块的配置文件是canonical croment的循环置换。
The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb{C}^n$ has a cluster algebra structure with Plücker relations giving rise to exchange relations. In this paper, we study indecomposable modules of the corresponding Grassmannian cluster categories ${\rm CM}(B_{k,n})$. Jensen, King, and Su have associated a Kac-Moody root system $J_{k,n}$ to ${\rm CM}(B_{k,n})$ and shown that in the finite types, rigid indecomposable modules correspond to roots. In general, the link between the category ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysterious and it is an open question whether indecomposables always give roots. In this paper, we provide evidence for this association in the infinite types: we show that every indecomposable rank 2 module corresponds to a root of the associated root system. We also show that indecomposable rank 3 modules in ${\rm CM}(B_{3,n})$ all give rise to roots of $J_{3,n}$. For the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to real roots, we show that their underlying profiles are cyclic permutations of a certain canonical one. We also characterize the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to imaginary roots. By proving that there are exactly 225 profiles of rigid indecomposable rank 3 modules in ${\rm CM}(B_{3,9})$ we confirm the link between the Grassmannian cluster category and the associated root system in this case. We conjecture that the profile of any rigid indecomposable module in ${\rm CM}(B_{k,n})$ corresponding to a real root is a cyclic permutation of a canonical profile.
