Asymptotic theory of characters of the symmetric group
Asymptotic theory of characters of the symmetric group
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对称群特征的渐近理论
DOI：
10.1007/bf01106153
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发表时间：
1981
影响因子：
0.4
通讯作者：
S. Kerov
中科院分区：
文献类型：
--
作者：
A. Vershik;S. Kerov
本文给出了无限对称群即可数集有限排列群6co的有限和半有限特征。 6~的有限特征首先在Thoma的重要论文中描述[12]。我们的方法基于其他思想，即从有限对称群 6~ 到极限。借助此方法，我们还可以获得有关大 n 的字符 ~ 的行为的信息。本文的内容与致力于 6~ o 组的作者的其他论文密切相关：在[16]中，我们给出了 II1 型因子表示的实现，在调查[17]中，我们使用 K0 的字符列表和一般属性计算了 ~ 的 K0 函子；在[15]中给出了Piancherel测度的渐近行为。在更广泛的背景下，群6co是局部有限群理论中的一个非平凡的实验模型，它的群代数在局部半简单代数理论中最近得到了深入的研究[18, 19]。这些物体的表征理论才刚刚开始发展。这里的问题与经典理论中的问题已经不同，因为通常所有表示的类都是巨大的（群和代数是“狂野的”）。此外，即使标准化字符集也可能是巨大的。更值得注意的是，对于经典系列 6~、U (oo)、GL (oo、Fq) 的限制，字符允许明确的描述。
In this paper there are found the finite and semifinite characters of the infinite symmetric group, ie, the group 6co of finitary permutations of a countable set. The finite characters of 6~ were first described in the important paper of Thoma [12]. Our method is based on other ideas, viz., on passage to the limit from the finite symmetric groups 6~. With the help of this method we also get information about the behavior of the characters of~ for large n. The content of the present paper is closely connected with other papers of the authors devoted to the group 6~ o: in [16] we give a realization of factor-representations of type II1, in the survey [17] we calculate the K0-funetor for~, using the list of characters and general properties of K0; in [15] there is given the asymptotic behavior of the Piancherel measure for~.In a wider setting, the group 6co is a nontrivial experimental model in the theory of locally finite groups, and its group algebra, in the theory of locally semisimple algebras has been intensively studied recently [18, 19]. The theory of representations of these objects is only beginning to be developed. The problems here are different than in the classical theory already because as a rule the class of all representations is immense (the groups and algebras are" wild"). Moreover, even the set of normalized characters can be immense. All the more remarkable that for the limits of the classical series 6~, U (oo), GL (oo, Fq) the characters admit explicit description.