SOME COMBINATORIAL AND ALGEBRAIC PROPERTIES OF COXETER COMPLEXES AND TITS BUILDINGS
SOME COMBINATORIAL AND ALGEBRAIC PROPERTIES OF COXETER COMPLEXES AND TITS BUILDINGS
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DOI：
10.1016/0001-8708(84)90021-5
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发表时间：
1984-01-01
影响因子：
1.7
通讯作者：
BJORNER, A
中科院分区：
文献类型：
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作者：
BJORNER, A
与考克斯特群和具有BN对的群相关联的是某些组合结构，称为考克斯特复形和山雀建筑。他们的理论，这是密切相关的性质，这样的群体，已发展主要是由山雀[39，40 1。近年来，Cohen-Macaulav复形的一般理论已经形成，它将组合结构与环理论联系起来。这一发展的主要建筑师是Hochster [18]和Stanley（34-37 I.可壳复形的概念为有限情形下的Cohen-Macaulay复形提供了一种构造性的方法，并且有其自身的几个有趣的特征。本文将从可壳复形和科恩-麦考利复形的角度研究考克斯特复形和山雀建筑。从Solomon-Tits同调计算[31 1]通过Reisner的工作[24 1]得出有限建筑物是Cohen-Macaulay。然而，无论是建筑物的斯坦利-莱斯纳环还是它的可壳性似乎都没有被明确考虑过。事实证明，考克斯特复形和建筑物是非常自然的可壳性，可壳性提供了快速和基本的访问其基本环理论和拓扑性质。在环论方面，这主要意味着，对于每个有限群的李（分别）。Coxeter）类型与某些Cohen-Macaulay（分别）相关。Gorenstein）环k [x，，x1.. u，J/Z，该群作用于其上。该环在一定的自然参数系统上具有正则的无特征基。这里k是一个域，不定式xi双射地对应于极大抛物子群系统的左陪集，理想1由对应于不相交的陪集对的所有单项式xixj生成。考克斯特复合体和建筑物的拓扑性质更为人所知。本组合方法的主要优点是，它导致一个简单而统一的治疗，同时包括所有类型的选择子复合物以及。
With Coxeter groups and groups with a BN-pair are associated certain combinatorial structures known as Coxeter complexes and Tits buildings. Their theory, which is intimately related to the properties of such groups, has been developed mainly by Tits [39, 40 1. In recent years a general theory of Cohen-Macaulav complexes has taken shape, which relates combinatorial structures with ring theory. The main architects of this development are Hochster [18 (and Stanley (34-37 I. The notion of a shellable complex provides a constructive approach to Cohen-Macaulay complexes in the finite case and has several interesting features of its own. In this paper Coxeter complexes and Tits buildings will be studied from the point of view of shellable and Cohen-Macaulay complexes. It follows from the Solomon-Tits homology computation [31 1 via the work of Reisner [24 1 that a finite building is Cohen-Macaulay. However, neither the Stanley-Reisner ring of a building nor its shellability seem to have been explicitly considered before. It turns out that Coxeter complexes and buildings are very naturally shellable and that shellability provides quick and elementary access to their basic ring-theoretic and topological properties. On the ring-theoretic side this means mainly that with each finite group of Lie (resp. Coxeter) type is canonically associated a certain Cohen-Macaulay (resp. Gorenstein) ring k [x,, x1....,. u, J/Z on which the group acts. This ring is provided with a canonical characteristic-free basis over a certain natural system of parameters. Here k is a field, the indeterminates xi correspond bijectively to the left cosets of a system of maximal parabolic subgroups, and the ideal 1 is generated by all monomials xixj corresponding to pairs of disjoint such cosets. The topological properties of Coxeter complexes and buildings are better known. The principal advantage of the present combinatorial approach is that it leads to a simple and unified treatment which simultaneously includes all type-selected subcomplexes as well.