论文标题
在对称特殊多层代数中包含字符串模块的试管
Tubes Containing String Modules in Symmetric Special Multiserial Algebras
论文作者
论文摘要
对称特殊的多粒代数是代数，对应于带有方向的装饰超图，称为Brauer配置。在本文中，我们使用Brauer配置的组合学通过其Auslander-Reiten Quiver来了解对称特殊多层代数的模块类别。特别是，我们提供了在稳定的Auslander-Reiten Quiver中仅使用基础brauer配置中的信息来确定稳定的Auslander-Reiten Quiver中试管的存在和等级的方法。首先，我们在Brauer配置（称为绿色的“超漫游”（HyperWalk）围绕着绿色的构造，定义了一个组合步行，该配置概述了围绕Brauer图的绿色步行的现有概念。然后，周期性的绿色超巡回赛被证明与相应的对称特殊多种代数上某些类别的字符串模块的定期分辨率相对应。因此，周期性的绿色超巡回赛决定了稳定的奥斯兰德 - 雷氏箭袋中的某些类管，由步行时期决定了管的等级。最后，我们提供了对称特殊多层代数的其他等级的描述，这些级别不是由绿色的超曲线引起的，但仍包含嘴巴的弦模块。这包括对第二等级管嘴的字符串模块之间的扩展空间的明确描述。
Symmetric special multiserial algebras are algebras that correspond to decorated hypergraphs with orientation, called Brauer configurations. In this paper, we use the combinatorics of Brauer configurations to understand the module category of symmetric special multiserial algebras via their Auslander-Reiten quiver. In particular, we provide methods for determining the existence and ranks of tubes in the stable Auslander-Reiten quiver of symmetric special multiserial algebras using only the information from the underlying Brauer configuration. Firstly, we define a combinatorial walk around the Brauer configuration, called a Green `hyperwalk', which generalises the existing notion of a Green walk around a Brauer graph. Periodic Green hyperwalks are then shown to correspond to periodic projective resolutions of certain classes of string modules over the corresponding symmetric special multiserial algebra. Periodic Green hyperwalks thus determine certain classes of tubes in the stable Auslander-Reiten quiver, with the ranks of the tubes determined by the periods of the walks. Finally, we provide a description of additional rank two tubes in symmetric special multiserial algebras that do not arise from Green hyperwalks, but which nevertheless contain string modules at the mouth. This includes an explicit description of the space of extensions between string modules at the mouth of tubes of rank two.
