Multiplicity free subgroups of compact connected Lie groups
Multiplicity free subgroups of compact connected Lie groups
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紧连通李群的重数自由子群
DOI：
10.1007/bf01224637
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发表时间：
1976
影响因子：
0.6
通讯作者：
M. Krämer
中科院分区：
文献类型：
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作者：
M. Krämer
设G是紧李群。对于群(或李代数)的表示，我们通常指的是有限维复表示，除非我们另有明确说明。G的一个闭子群H称为重数有界子群，如果它具有下列性质：存在一个常数K>；0使得对G的任一不可约表示，对H的限制i~的所有不可约H-分支至多有K个重数，如果K=1，我们称H为重数子群。在[5]中，作者给出子群无重数的一个充要条件(文[5]中的定理1)。他们还讨论了多重性自由子群在物理学中的意义。[5]中的情况与早期的上帝标准密切相关([4]，13，科罗尔)。关于西奥尔的。从几何学的角度来看，这肯定是……正确的“条件。然而，为了分类的目的，它并不是那么可行。本文给出了紧连通李群的无重数次群的一种分类(见我们的定理)。事实证明，人们只得到这样的子群，这些子群经典地被认为是无重性的。我们的方法与[5]中的方法大不相同。为了进行分类，我们得到了闭子群H是重有界的一个必要条件(命题1)，该条件在后验证明也是充分的。证明命题1的方法可能有它自己的兴趣：通过对表示的维度、权重的重数等的渐近行为的一般估计，我们获得了一个足够强大的具体条件，以相当快地提供所需的分类。
Let G be a compact Lie group. By a representation of a group (or a Lie algebra) we always mean a finite dimensional complex representation unless we explicitly say otherwise. A closed subgroup H of G is called a multiplicity bounded subgroup if it has the following property: There is a constant K> 0 such that for any irreducible representation@ of G, all the irreducible H-components of the restriction@ I~ of@ to H have multiplicity at most K. If we can choose K= 1, we call H a multiplicity] ree subgroup. In [5] the authors give a necessary and sufficient condition for a subgroup to be multiplicity free (Theorem 1 in [5]). They also discuss the significance of multiplicity free subgroups in physics. The condition in [5] is closely related to an earlier criterion of Godement ([4], 13, Coroll. of Theor. 8) and is surely the,, right" condition from a geometrical point of view. For the purpose of classification however it is not so practicable. In the present paper we give a classification of the multiplicity free subg-{oups of compact connected Lie groups (see our Theorem). It turns out that one gets only such subgroups which are classically known to be multiplicity free. Our approach is quite different from that in [5]. To do the classification, we derive a necessary condition for the closed subgroup H to be multiplicity bounded (Proposition 1) which a posteriori turns out to be also sufficient. The way of proving Proposition 1 may have an interest of its own: By means of general estimations on the,, asymptotical" behaviour of dimensions of representations, of multiplicities of weights and so on, we obtain a concrete condition which is powerful enough to deliver the desired classification rather quickly.
DOI：
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发表时间：
2005
期刊：
影响因子：
--
作者：
T. Kobayashi;T. Oshima
通讯作者：
T. Oshima