Asymptotic Results on Modular Representations of Symmetric Groups and Almost Simple Modular Group Algebras
Asymptotic Results on Modular Representations of Symmetric Groups and Almost Simple Modular Group Algebras
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对称群模表示和几乎简单模群代数的渐近结果
DOI：
10.1006/jabr.1999.7923
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发表时间：
1999
影响因子：
0.9
通讯作者：
A. Zalesskiĭ
中科院分区：
文献类型：
--
作者：
A. Baranov;A. Kleshchev;A. Zalesskiĭ
事实上，这是一个关于无限单群的问题，因为G是有限的很容易，并且因为一个非平凡的正规子群会产生一个不同于增广理想的非平凡理想。还要注意的是，这个问题很容易简化为关于什么时候增广理想像环一样简单的问题。第一个有趣的几乎单FG群类是在[3]中发现的。这个类是相当奇特的，包含像通用霍尔群和代数闭群这样的群。对于局部有限群G，最近利用有限群的表示理论取得了实质性进展，见[18，19，16]等。表示论方法将理想问题转化为有限群表示的渐近行为的某些问题，这些问题通常是独立的。该方法是更有效的领域的特征为零的普通表示理论是更好地阐述比模块化理论。本文致力于模块化方面的理论。我们的一个主要结果涉及的情况下，G是一个直接限制的交错群和F是任何领域的特征p > 2。我们注意到在[18]中考虑了charF = 0的情况。设N是自然数的集合。分别用Alt（Ω）和Sym（Ω）（或者简单地说An和Sym n）表示集合Ω上的交错群和对称群，|Ω| = n.设Alt（Ω1）<$Alt（Ω2）<$··<$Alt（Ωi）<$。. .（一）
In fact, this is a question about infinite simple groups because it is easy for G finite, and because a non-trivial normal subgroup gives rise to a non-trivial ideal different from the augmentation ideal. Also note that the problem reduces easily to the question on when the augmentation ideal is simple as a ring. The first interesting class of groups with almost simple FG was discovered in [3]. This class is rather exotic and contains groups like the universal Hall group and algebraically closed groups. For locally finite groups G a substantial progress was achieved recently by using representation theory of finite groups, see [18, 19, 16], and others. The representation theory approach transforms the problems on ideals to certain problems on asymptotic behavior of representations of finite groups, which are often of independent interest. The method is more effective over fields of characteristic zero as the theory of ordinary representations is much better elaborated than the modular theory. This paper is devoted to the modular aspect of the theory. One of our main results deals with the case where G is a direct limit of alternating groups and F is any field of characteristic p > 2. We note that the case where charF = 0 was considered in [18]. Let N be the set of natural numbers. Denote by Alt(Ω) and Sym(Ω) (or, simply, An and Σn) the alternating and symmetric groups, respectively, on a set Ω with |Ω| = n. Let Alt(Ω1) ⊂ Alt(Ω2) ⊂ · · · ⊂ Alt(Ωi) ⊂ . . . (1)