The Bruhat order of the symmetric group is lexicographically shellable
The Bruhat order of the symmetric group is lexicographically shellable
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对称群的 Bruhat 阶按字典顺序可脱壳
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发表时间：
1981
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通讯作者：
Paul H. Edelman
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文献类型：
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作者：
Paul H. Edelman
本文给出了对称群S的Bruhat序是字典序可壳的，因而是Cohen-Macaulay序的初等证明.利用Verma的一个定理，我们得到了一个推论，即S的链的单纯复形LA（Sn）是一个在维数为2的球面三角剖分上的双锥。我们将使用Bjorner [2]的符号和术语。一个有限偏序集P称为有界的，如果它有一个最大元和一个最小元，分别记为I和0。它被称为纯的，如果它的所有极大链是相同的长度，它是分次的，如果它是有界的和纯的。P的秩是极大链的长度。分次偏序集P的元素x有一个定义明确的秩p（x），它等于P中从0到x的不可加细链的长度。如果P是有界的，设P是偏序集P {0，1}。偏序集P的阶复形/（P）是P中所有链的单纯复形。如果/（P）是可壳的，则称偏序集是可壳的。关于可壳复形的定义，请参见[2]或[4]。类似地，P称为Cohen-Macaulay，如果f（P）为。关于Cohen-Macaulay复形的定义和意义，见[1]、[2]或[6]。设C（P）是覆盖关系的集合
In this note we present an elementary proof that the Bruhat order of the symmetric group S, is lexicographically shellable and hence Cohen-Macaulay. Using a theorem of Verma we obtain as a corollary that LA(Sn), the simplicial complex of chains of S, is a double cone over a triangulation of a sphere of dimension () 2. We will employ the notation and terminology of Bjorner [2]. A finite poset P is said to be bounded if it has a maximum and a minimum element, denoted I and 0 respectively. It is calledpure if all of its maximal chains are the same length and it is graded if it is both bounded and pure. The rank of P is the length of a maximal chain. An element x of a graded poset P has a well-defined rank p(x) equal to the length of an unrefinable chain from 0 to x in P. If P is bounded let P be the poset P {0, 1}. The order complex /(P) of a poset P is the simplicial complex of all chains in P. A poset is said to be shellable if /(P) is shellable. For the definition of a shellable complex see [2] or [4]. Similarly P is called Cohen-Macaulay if /(P) is. See [1], [2] or [6] for the definition and significance of a Cohen-Macaulay complex. Let C(P) be the set of covering relations