Abstract

For an element w of a simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of the module category over the preprojective algebra of Dynkin type. This paper studies categorical properties of $\mathcal{F}(w)$ using the root system. We show that simple objects in $\mathcal{F}(w)$ bijectively correspond to Bruhat inversion roots of w, and obtain a combinatorial criterion for $\mathcal{F}(w)$ to satisfy the Jordan-Hölder property (JHP). 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.

摘要

对于简单定位的Weyl组的元素w，Buan-Iyama-Reiten-Scott定义了一个子类别$\mathcal{F}（w）$Dynkin类型的预投影代数上的模数类别的集合。本文研究了分类的性质$\mathcal{F}（w）$使用根系统。我们展示了简单的对象$\mathcal{F}（w）$双射地对应于w的Bruhat反演根，并获得w的组合准则$\mathcal{F}（w）$满足约旦·霍尔德（JHP）的财产。对于A型案例，我们给出了简单对象的图解结构，并表明（JHP）可以通过类似森林的排列来表征，这是Bousquet-Mélou和Butler在研究Schubert品种时引入的。