Integral expression of some indecomposable characters of the infinite symmetric group in terms of irreducible representations
Integral expression of some indecomposable characters of the infinite symmetric group in terms of irreducible representations
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无限对称群的一些不可分解特征的不可约表示的积分表达
DOI：
10.1007/bf01446899
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发表时间：
1990
影响因子：
1.4
通讯作者：
N. Obata
中科院分区：
文献类型：
--
作者：
N. Obata
根据定义，无限对称群 ~ o 是自然数集 l~={1, 2....} 的所有有限排列的离散群。自 Murray 和 yon Neumann [6] 以来，它经常被引用为 ICC 群的典型例子，因此也是非 I 型群的典型例子。因此，无限对称群应该包含许多在 I 型群中未观察到的有趣特征，并且已成为表示论的一个不可忽视的新分支。 Thoma [12] 的重要论文提出了~ oo 表示论的第一个系统方法，他给出了~| 的不可分解特征的完整列表。即，所有 II 因子表示的参数化。 Vershik 和 Kerov [4, 13-16] 对此进行了大量讨论。至于 oo 的不可约表示，Lieberman [5] 确定了所有相对于特定拓扑连续的表示（另见 Ol'shanskii [11]）。然而，拓扑结构非常弱，以至于他的列表仅包含可数无限个不可约表示，根本不足以讨论与因子表示的联系。受这些工作的激励，作者 [7, 8] 开始了一项新的尝试，以获得更广泛的具有离散拓扑的不可约表示。 Hirai [2, 3] 高度推广了该方法，以获得一类非常大的不可约表示。此外，他还在花环产品研究中研究了一些新现象。在本文中，我们的目标是揭示不可分解的字符和不可约的表示之间的联系。更准确地说，我们将证明某些不可分解的特征允许用正定函数来表示积分表达式，这些表达式生成 ~ o 的不可约表示。这些不可约表示是由“无限Young子群”通过归纳而构造出来的，并形成平井股票的特殊子类[3]。此外，带有概率测度的“无限Young画面”空间在积分表达中起着重要作用。主要结果如定理3.2所示。
The infinite symmetric group~ o is by definition the discrete group of all finite permutations of the set of natural numbers l~={1, 2....}. Since Murray and yon Neumann [6] it has been often quoted as a typical example of ICC-groups and hence of groups of non-type I. For that reason the infinite symmetric group should involve a number of interesting features which are not observed in groups of type I and has become a new branch of representation theory to be reckoned with. The first systematic approach to representation theory of~ oo was made in the important paper by Thoma [12] who gave a complete list of indecomposable characters of~| namely, a parametrization of all IIl-factor representations of~. Further related topics in this connection have been considerably discussed by Vershik and Kerov [4, 13-16]. As to irreducible representations of~ oo, Lieberman [5] determined all of those which are continuous with respect to a particular topology (see also Ol'shanskii [11]). However, the topology is so weak that his list consists of only countably infinite number of irreducible representations and is not enough to discuss connection with factor representations at all. Motivated by these works, the author [7, 8] started on a new attempt to obtain a wider class of irreducible representations of~ equipped with the discrete topology. The method employed there is highly generalized by Hirai [2, 3] to obtain a very large class of irreducible representations. In addition he has investigated a number of new phenomena in the study of wreath products as well. In this paper we aim at revealing a connection between indecomposable characters and irreducible representations of~ o. More precisely, we shall show that certain indecomposable characters admit integral expressions in terms of positive definite functions which generate irreducible representations of~ o. These irreducible representations are constructed from" infinite Young subgroups" by means of inducing up and form a special subclass of Hirai's stock [3]. Furthermore, the space of" infinite Young tableaux" equipped with a probability measure plays an important role in the integral expression. The main result is stated in Theorem 3.2.