A Two‐Sided Cell in an Affine Weyl Group
A Two‐Sided Cell in an Affine Weyl Group
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DOI：
10.1112/jlms/s2-36.3.407
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发表时间：
1987-12
影响因子：
1.2
通讯作者：
Jian-yi Shi
中科院分区：
文献类型：
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作者：
Jian-yi Shi
由于An的任何双边胞腔都是An'的某些RL-等价类的并集，这意味着A的双边胞腔的数目是有限的，并且小于或等于n的划分数。到目前为止，这里的所有结果都是用我们自己的初等方法得到的，不依赖于交上同调理论的知识。但现在我们将使用最近的结果Lusztig，即定理1.5。3 [L12]证明了An的任何RL-等价类实际上是An的双边胞腔，然后我们可以确定An的双边胞腔的确切数目，它等于n的划分数。Lusztig的这一结果来自于相交上同调的深层理论。我们知道Lusztig [L13]用与我们类似的方法证明了本章的主要结果。但我们独立地做这件事。为了说明我们的结果，我们首先给出一些引理。符号y？w意味着y在An'的相同的双侧单元格中。
Since any two-sided cell of An is a union of some RL-equivalence classes of An'this implies that the number of two-sided cells of A is finite and is less than or equal n to the number of partitions of n. So far, all the results here have been obtained by our own elementary methods and do not rely on the knowledge of the intersection cohomology theory. But now we shall use a recent result of Lusztig, ie Theorem 1.5. 3 [L12] in Chapter 1 to prove that any RL-equivalence class of An is actually a two-sided cell of An'Then we can determine the exact number of two-sided cells of An which is equal to the number of partitions of n. This result of Lusztig comes from the deep theory of intersection cohomology. We understand that Lusztig [L13] showed the main result of this chapter in the similar way as we do. But we do this independently. To show our result, we shall first give some lemmas. The notation y? w means that y, ware in the same two-sided cell of An'