Coxeter groups and quiver representations
Coxeter groups and quiver representations
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Coxeter 组和箭袋表示
DOI：
10.1090/conm/716/14429
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
H. Thomas
中科院分区：
文献类型：
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作者：
H. Thomas
在这篇简短的笔记中，我展示了Coxeter群与非线性表示的相关性。我讨论（1）真实的和虚根，（2）反射函子，（3）扭转自由类和c-排序元素。前两个主题是经典的，而第三个是最近的发展。我表明，挠自由类在RepQ包含1000多个不可分解的双射对应的C-可排序的元素在相应的Weyl组。这一点首先由英格尔斯和托马斯在Dynkin类型中确立; Amiot、Iyama、Reiten和Todorov一般地表明了这一点。本注中的证明是基本的，基本上遵循Ingalls和托马斯的论证，但没有假设Q是Dynkin。
In this expository note, I showcase the relevance of Coxeter groups to quiver representations. I discuss (1) real and imaginary roots, (2) reflection functors, and (3) torsion free classes and c-sortable elements. The first two topics are classical, while the third is a more recent development. I show that torsion free classes in rep Q containing finitely many indecomposables correspond bijectively to c-sortable elements in the corresponding Weyl group. This was first established in Dynkin type by Ingalls and Thomas; it was shown in general by Amiot, Iyama, Reiten, and Todorov. The proof in this note is elementary, essentially following the argument of Ingalls and Thomas, but without the assumption that Q is Dynkin.