The Catalan combinatorics of the hereditary artin algebras
The Catalan combinatorics of the hereditary artin algebras
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遗传性代数的加泰罗尼亚组合数学
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发表时间：
2015
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通讯作者：
C. Ringel
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作者：
C. Ringel
本文利用遗传Artin代数H的表示理论，研究了模H中例外子范畴集P的范畴化问题。这种分类是由于英格尔斯和托马斯，以及随后的论文由伊古萨和希夫勒。
出发点是一个完善的经典倾斜理论的模H，取代扭转对扭转三元组，从而把它纳入王国的稳定性理论。模H中的挠对很好地对应于例外子范畴的垂直对，并且在模和子范畴的集合之间存在大量的双射，即英格斯-托马斯双射。
如果H是表示有限的，我们可以看对应的模或子范畴的数目。这样的Dynkin函数（它们附加到Dynkin图上的整数）在第1章中显示。在一个神秘的方式，许多Dynkin函数可以描述使用指数的外尔群。根据Shapiro和Kostant，指数由根偏序集的高度划分给出。Abe-Barakat-Cuntz-Hoge-Terao最近的一个结果允许归纳地确定它们，在根偏序集中的理想链中向上，查看相应的超平面排列。
第四章讨论了Dynkin型A线性定向的情形。这里P被确定为格NC的非交叉分区介绍了Kreweras（现在的重要工具，在几个部分的数学，例如在自由概率论）。我们回顾了NC中与极大链有关的一些经典问题：标号树的计数以及停车函数的计数。Dynkin情形A的组合学就是Catalan数的组合学;在附录中，我们讨论了经典Catalan组合学的性质。
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler.
Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories and there is a wealth of bijections, the Ingalls-Thomas bijections, between sets of modules and subcategories.
If H is representation-finite, one may look at the corresponding numbers of modules or subcategories. Such Dynkin functions (they attach to a Dynkin diagram an integer) are displayed in chapter 1. In a mysterious way, many Dynkin functions can be described using the exponents of the Weyl group. According to Shapiro and Kostant, the exponents are given by the height partition of the root poset. A recent result of Abe-Barakat-Cuntz-Hoge-Terao allows to determine them inductively, going up in a chain of ideals in the root poset, looking at the corresponding hyperplane arrangements.
Chapter 4 deals with the case of the linearly oriented quiver of Dynkin type A. Here P is identified with the lattice NC of non-crossing partitions as introduced by Kreweras (now an important tool in several parts of mathematics, for example in free probability theory). We review some classical problems which are related to the maximal chains in NC: to count labeled trees as well as parking functions. The combinatorics of the Dynkin case A is just the combinatorics of the Catalan numbers; in an appendix, we discuss the nature of classical Catalan combinatorics.