REPRESENTATION THEORY OF SYMMETRIC GROUPS AND THEIR DOUBLE COVERS
REPRESENTATION THEORY OF SYMMETRIC GROUPS AND THEIR DOUBLE COVERS
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对称群及其双覆盖的表示论
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
A. Kleshchev
中科院分区：
文献类型：
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作者：
Jonathan Brundan;A. Kleshchev
在本文中，我们将概述最近几年出现的对称群的 p 模表示论及其双重覆盖的新李理论方法。事实上，这里有两个平行的理论：一种是关于涉及 A p−1 类型的仿射 Kac-Moody 代数的对称群 Sn，另一种是关于涉及 A p−1 类型的扭曲代数的双覆盖 Ŝn 的理论。就 Sn 本身而言，该理论尤其由 Kleshchev [19]、Lascoux-Leclerc-Thibon [21]、Ariki [1] 和 Grojnowski [9] 发展，而在 Sergeev [35, 36] 和 Nazarov [30, 31] 对 C 取得重要进展之后，双重覆盖在 [4] 中首次沿着 [9] 的思路进行处理。这两种理论的核心最引人注目的结果之一是用柏原晶体图对相应仿射李代数的基本模块的模块分支图进行了明确的描述。请注意，所描述的结果只是更大图景的一部分：分圆和仿射 Hecke 代数以及它们的扭曲类似物，分圆和仿射 Hecke-Clifford 超代数也有类似的结果。然而，我们在这里将尝试仅提出理论中适用于对称群的部分，因为它最适用于有限群理论。
In this article we will give an overview of the new Lie theoretic approach to the p-modular representation theory of the symmetric groups and their double covers that has emerged in the last few years. There are in fact two parallel theories here: one for the symmetric groups Sn involving the affine Kac-Moody algebra of type A p−1, and one for their double covers Ŝn involving the twisted algebra of type A p−1. In the case of Sn itself, the theory has been developed especially by Kleshchev [19], Lascoux-Leclerc-Thibon [21], Ariki [1] and Grojnowski [9], while the double covers are treated for the first time in [4] along the lines of [9], after the important progress made over C by Sergeev [35, 36] and Nazarov [30, 31]. One of the most striking results at the heart of both of the theories is the explicit description of the modular branching graphs in terms of Kashiwara’s crystal graph for the basic module of the corresponding affine Lie algebra. Note that the results described are just a part of a larger picture: there are analogous results for the cyclotomic and affine Hecke algebras, and their twisted analogues, the cyclotomic and affine Hecke-Clifford superalgebras. However we will try here to bring out only those parts of the theory that apply to the symmetric group, since that is the most applicable to finite group theory.