On Some Simple Constituents of the Specht Modules of the Symmetric Groups
On Some Simple Constituents of the Specht Modules of the Symmetric Groups
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关于对称群Spect模的一些简单成分
DOI：
10.1006/jabr.2001.8872
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发表时间：
2001
影响因子：
0.9
通讯作者：
Y. Tsushima
中科院分区：
文献类型：
--
作者：
Y. Tsushima
设Sn是n个字母上的对称群，L是域.给定整数n的一个划分λ，我们有一个对应于λ的LSn-模S，称为Specht模。如果L的特征为零，则它们的集合λ穿过n的划分，形成非同构的简单LSn-模的代表的全集。如果L有素特征，比如p，它们不一定是单的。然而，如果划分λ是p-正则的，则S的标头D是单的，并且当λ穿过n的p-正则划分时，它们覆盖所有非同构的单模。关于Specht模块的主要问题之一是要了解它们的简单组成部分。特别地，我们想从λ的知识中知道p-正则划分μ使得D是S的一个分支。关于这一点，卡特和佩恩的定理（见[1，定理，第425页]）往往是强大的。另一方面，Jantzen-Schaper定理（见定理1）告诉我们，如果我们知道对应于严格支配λ的划分的Specht模的分解数，我们就可以确定S的所有简单成分。他们的方法涉及对James和Murphy [5]引入的划分λ的某些操作，其中每一个操作都被粗略地解释为在对应于λ的Young图上去除边缘钩，然后添加。我们称每个所得的划分为λ的一个分支，如果λ = μ或μ是通过从λ开始连续分支得到的，则记为λ → μ。Jantzen-Schaper定理特别指出，如果D是S的一个分支，则λ → μ。当然匡威就不是了
Let Sn be the symmetric group on n letters and L a field. Given a partition λ of the integer n, we have an LSn-module S called the Specht module corresponding to λ. If the characteristic of L is zero, the set of them as λ runs through the partitions of n forms a full set of representatives of the non-isomorphic simple LSn-modules. If L has prime characteristic, say p, they are not necessarily simple. However, if the partition λ is p-regular, the head of S, denoted by D, is simple and they cover all the non-isomorphic simple modules as λ runs through the p-regular partitions of n. One of the main concerns about the Specht modules is to have information about the simple constituents of them. Especially, we want to know about p-regular partitions μ such that D is a constituent of S from the knowledge of λ. Concerning this, Carter and Payne’s theorem (see [1, Theorem, p. 425]) is often powerful. On the other hand the Jantzen– Schaper theorem (see Theorem 1) tells us that if we know the decomposition numbers for the Specht modules which correspond to partitions which strictly dominate λ, we can determine all the simple constituents of S. Their methods involve certain operations on the partitions λ introduced by James and Murphy [5], each of which is roughly interpreted as a rim hook removal followed by addition on the Young diagram corresponding to λ. We shall call each of the resulting partitions a branch of λ and write λ → μ if λ = μ or μ is obtained by making branches successively beginning with λ. The Jantzen–Schaper theorem tells in particular that if D is a constituent of S, it follows that λ → μ. Of course the converse is not