The decomposition into cells of the affine Weyl groups of type A
The decomposition into cells of the affine Weyl groups of type A
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A 型仿射 Weyl 群分解为元胞
DOI：
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发表时间：
1984
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影响因子：
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通讯作者：
Jian
中科院分区：
文献类型：
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作者：
Jian
在文献[1]中，Kazhdan和Lusztig引入了Coxeter群W的W-图的概念。特别是，它们定义了左、右和双侧细胞。这些W-图在表示论中起着重要的作用。然而，Kazhdan和Lusztig给出的计算这些细胞的算法非常复杂。这些细胞只在极少数情况下被计算出来。本文给出了An- 1 > 2型仿射Weyl群An的所有左胞腔、右胞腔和双边胞腔。我们的主要结果表明，每一个左（resp。A n的（右）胞腔决定n的一个分拆，比如λ，且λ的特征在于λ-tabloid和它的广义右（分别为.左）T不变。An的双边胞腔集合与n的分拆集合An之间存在一一对应。左数（左数）右）对应于给定划分λ的单元等于n l / m j = 1 u j l，其中{u 1 >.> um}是λ的对偶划分。An中的每个双边胞腔也是An的一个RL-等价类，并且是一个连通集。每一个左（左）（右）单元格是一个最大的左（或左）。设P是An的任意一个同构于对称群Sn-的真标准抛物子群，则P与An的每一个双边胞腔的交都是非空的，并且是P的双边胞腔。（右）A n的细胞。这些结果大多由Lustig [2]、[3]所证实。
In [1], Kazhdan and Lusztig introduce the concept of a W-graph for a Coxeter group W. In particular, they define left, right and two-sided cells. These W-graphs play an important role in the representation theory. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated. These cells have been worked out only in a very few cases. In the present thesis, we shall find all the left, right and two-sided cells in the affine Weyl group A n of type A n - 1 > 2. Our main results show that each left (resp. right) cell of A n determines a partition, say λ of n and, is characterized by a λ-tabloid and also by its generalized right (resp. left ) T-invariant. There exists a one-to-one correspondence between the set of two-sided cells of A n and the set A n of partitions of n. The number of left (resp. right) cells corresponding to a given partition λ ϵ A n is equal to n l / m п j = 1 u j l , where {u 1 > … > um} is the dual partition of λ. Each two- sided cell in A n is also an RL-equivalence class of A n and is a connected set. Each left (resp. right) cell in A n is a maximal left (resp. right) connected component in the two-sided cell of A n containing it. Let P be any proper standard parabolic subgroup of A n isomorphic to the symmetric group S n - then the intersection of P with each two-sided cell of A n is non-empty and is just a two-sided cell of P. The intersection of P with each left (rasp, right) cell of A n is either empty or a left (resp. right) cell of A n. Most of these results were conjectured by Lusstig [2], [3].