The Schur index of projective characters of symmetric and alternating groups
The Schur index of projective characters of symmetric and alternating groups
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对称群和交替群投影特征的 Schur 指数
DOI：
10.2307/2946564
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发表时间：
1992
影响因子：
4.9
通讯作者：
A. Turull
中科院分区：
文献类型：
--
作者：
A. Turull
本文件回答了一个自然的问题，出现时，一套并排的两个主要贡献的伊赛舒尔理论的群表示。在一篇不朽的论文[13]中，舒尔描述了对称群Sn和交错群An）对所有n的射影特征标。他计算了Sn和An）对所有n的表示群，并计算了这些表示群的特征标表，从而描述了Sn和An的射影特征标。在另一篇开创性的论文[12]中，舒尔定义并研究了现在所谓的舒尔指数。若4f是某个有限群G的不可约特征标，K是特征为零的域，则qf关于K的Schur指标mK（G）是qf在KG-模所提供的特征标中的最小正重数。问题是：Sn和An的表示群的特征标的Schur指数是多少？本文给出了一个规则，它描述了Sn和An）的表示群的每个不可约特征标的Schur指标。大量的研究致力于舒尔指数。努力已被定向到找到它的一般属性和计算它的重要类别的例子。例如，Benard [1]证明了E6、E7和E8型Weyl群的每个特征标的Schur指数都是1。在[2]中，Feit给出了计算许多特殊群的Schur指数的简短证明，包括E6，E7和E8型Benard Weyl群，以及零星单群的表示群。其他作者研究了李型群的舒尔指数;例如见[4]或[10]。有各种方法可以给出Schur指数的上界，但当它们大于1时，通常很难找到它们的值。因此，对于李型群，我们可以证明它们的舒尔指数在许多情况下至多为2，在某些情况下至多为1，但是对于哪些特征标它是1，哪些是2的问题还没有得到一致的回答。
The present paper answers a natural question that arises when one sets side by side two major contributions of Issai Schur to the theory of group representations. In a monumental paper [13], Schur described the projective characters of the symmetric group Sn and the alternating group An) for all n. He calculated the representation groups of Sn and An) for all n, and calculated the character table of these representation groups, thereby describing the projective characters of Sn and An. In another seminal paper [12], Schur defined and studied what is now called the Schur index. If 4f is an irreducible character of some finite group G and K is a field of characteristic zero, the Schur index mK(G) of qf with respect to K is the least positive multiplicity of q/ in a character afforded by a KG-module. The question is: What are the Schur indices of the characters of the representation groups of Sn and An? We give here a rule that describes the Schur index of every irreducible character of the representation groups of Sn and An) for all n. A substantial amount of research has been devoted to the Schur index. Efforts have been directed both to finding its general properties and to calculating it for important classes of examples. For example, Benard [1] showed that the Schur index of every character of the Weyl groups of type E6, E7 and E8 is one. In [2], Feit gives short proofs for the calculation of the Schur index for many specific groups, including Benard's Weyl groups of type E6, E7 and E8, and the representation groups of the sporadic simple groups. Other authors have studied the Schur index for groups of Lie type; see for example [4] or [10]. Various methods are available to give upper bounds for the Schur indices, but it is usually harder to find their value when they are larger than one. Hence, for the groups of Lie type, one can show that their Schur indices are at most two in many cases and at most one in some cases, but the question of exactly for what characters it is one and for which it is two has not been consistently answered.