论文标题
韦尔组和无扭转类的bruhat反演
Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
论文作者
论文摘要
对于简单的Weyl组的元素$ W $，Buan-iyama-reiten-Scott定义了一个模块类别的子类别的子类别$ \ MATHCAL {F}（w）$，而不是Dynkin类型的预选代数。本文旨在通过与根系的连接来研究$ \ Mathcal {f}（w）$的分类属性。我们表明，通过服用尺寸向量，$ \ Mathcal {f}（w）中的简单对象都对应于$ w $的bruhat反转根。作为应用程序，我们获得了$ \ Mathcal {f}（w）$的组合标准，以满足Jordan-Hölder属性（JHP）。为了实现这一目标，我们开发了一种方法，通过使用与最大绿色序列相关的砖序列，在无扭转类中找到简单对象。对于A型案例，我们给出了简单物体的图形结构，并表明（JHP）可以通过Bousquet-Mélou和Butler在Schubert品种研究中引入的类似森林的置换来表征。
For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of a module category over a preprojective algebra of Dynkin type. This paper aims at studying categorical properties of $\mathcal{F}(w)$ via its connection with the root system. We show that by taking dimension vectors, simple objects in $\mathcal{F}(w)$ bijectively correspond to Bruhat inversion roots of $w$. As an application, we obtain a combinatorial criterion for $\mathcal{F}(w)$ to satisfy the Jordan-Hölder property (JHP). To achieve this, we develop a method to find simple objects in a general torsion-free class by using a brick sequence associated to a maximal green sequence of it. For type A case, we give a diagrammatic construction of simple objects, and show that (JHP) can be characterized via a forest-like permutation, introduced by Bousquet-Mélou and Butler in the study of Schubert varieties.
