The 2-Blocks of the Covering Groups of the Symmetric Groups
The 2-Blocks of the Covering Groups of the Symmetric Groups
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对称群覆盖群的2-分块
DOI：
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发表时间：
1997
期刊：
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通讯作者：
J. B. Olsson
中科院分区：
文献类型：
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作者：
C. Bessenrodt;J. B. Olsson
摘要令n是n次有限对称群S n的双覆盖，即，n有一个中心对合z，使得n/z S n。根据它的核中是否有z，将n的不可约特征标称为普通特征标或自旋特征标。本文的目的是确定自旋特征标在2-块中的分布。这里采用的方法与以前处理这类问题的方法有本质的不同。我们还讨论了我们的主要结果的分解数的一些后果。证明了James关于对称群分解数的一个类似结果，并推广了Benson的一个定理[Ben，Theorem 1.2].在第1节中，我们提出了我们的结果的背景，并给出了一些注释。在第二节中，我们给出了2-块中自旋特征标个数的一个显式公式。我们还证明了一个结果的重量块含有一个给定的非自缔合自旋字符，这将是重要的证明我们的定理的自旋字符的2块分布。第3节介绍了第4节和第5节中使用的一些基本组合概念。第四节证明了关于给定2-块的自旋特征标的定理，第五节给出了关于分解数的结果。
Abstract Let Ŝ n be a double cover of the finite symmetric group S n of degree n , i.e., Ŝ n has a central involution z such that Ŝ n /⦠ z ⦔≃ S n . An irreducible character of Ŝ n is called ordinary or spin according to whether it has z in its kernel or not. The purpose of this paper is to determine the distribution of the spin characters of Ŝ n into 2-blocks. The methods applied here are essentially different from those applied to previous questions of this type. We also discuss some consequences of our main result for the decomposition numbers. An analogue of James' well-known result for the decomposition numbers of the symmetric groups is proved, providing also a generalization of a theorem of Benson [Ben, Theorem 1.2]. In Section 1 we present the background for our results and give some preliminaries. In Section 2 we give an explicit formula for the number of spin characters in a 2-block. We also prove a result about the weight of a block containing a given non-self-associate spin character which will be important for the proof of our theorem on the 2-block distribution of spin characters. Section 3 presents some fundamental combinatorial concepts used in Sections 4 and 5. The theorem concerning the spin characters in a given 2-block is proved in Section 4, and in Section 5 we present our results on the decomposition numbers.