On the action of the symmetric group on the Free Lie Algebra and the partition lattice
On the action of the symmetric group on the Free Lie Algebra and the partition lattice
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对称群对自由李代数和配分格的作用
DOI：
10.1016/0097-3165(90)90050-7
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
H. Barcelo
中科院分区：
文献类型：
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作者：
H. Barcelo
字母A上的自由李代数，这里记为LIE[A]，是包含字母的A-词的线性跨度的最小子空间，并且在括号运算[f，g] =fg−gf下闭合。置换σ通过用σi替换每个出现的字母来作用于单词。这个动作线性地延伸到LIE[A]。我们在这里关心的作用的对称群Snon的子空间的李[A]这是线性跨度括号的话是排列的字母表。从Hanlon、Stanley和Joyal的工作中可以看出，这个作用和Snon的作用是划分格的顶同调，它诱导了类似的表示（直到具有交替特征的张量）。根据Garsia和Stanton的工作，对同调的作用类似于对斯坦利-赖斯纳环的适当定义的顶部的作用。在本文中，我们得到了一个直接的组合证明，这三个行动的相似性，通过选择在这三个空间中的每一个自然基和比较矩阵对应的简单的反射。
The Free Lie Algebra over an alphabetA, denoted here by LIE[A], is the smallest subspace of the linear span of theA-words which contains the letters and is closed under the bracket operation [f,g] =fg−gf. A permutation σ acts on words by replacing each occurrence of the letteraibyaσi. This action linearly extends to LIE[A]. We are concerned here with the action of the symmetric groupSnon the subspace of LIE[A] which is the linear span of bracketings of words which are permutations of the letters of the alphabet. It follows from the work of Hanlon, Stanley, and Joyal that this action and the action ofSnon the top homology of the partition latticeΠninduce similar representations (up to tensoring with the alternating character). It follows from the work of Garsia and Stanton that the action on the homology is similar to the action on a suitably defined top portion of the Stanley-Reisner ring. In this paper we derive a direct combinatorial proof of the similarity of these three actions by choosing natural bases in each of these three spaces and comparing the matrices corresponding to the simple reflections.