A Generalization of the Littlewood-Richardson Rule
A Generalization of the Littlewood-Richardson Rule
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DOI：
10.1016/0021-8693(90)90086-4
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发表时间：
1990-05
影响因子：
0.9
通讯作者：
P. Littelmann
中科院分区：
文献类型：
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作者：
P. Littelmann
对于一般线性群 GZ，Littlewood-Richardson 规则（参见 [18, 193）给出了计算两个不可约 G/,-表示的张量积的分解的方法。本文的目的是为所有类型为 A,, B,, C,, D,, G2 和 E, 的简单、单连通代数群给出这条规则的推广。我们还获得了 F4、E 和 ES 类型的 G 的部分结果。最后三种情况的限制来自于这样的事实：为了分解规则的制定，我们需要标准杨氏表的概念。 Seshadri、Lakshmibai、Musili 和 Rajeswari 在一系列文章中提出了这样的概念（参见 [ll, 13, 14, 16]），但尚未适用于最后三个特殊群体的所有表示。 Seshadri 等人提出的年轻画面概念的优势在于：是它独立于组的类型。为了方便不习惯这个概念的读者，我们首先使用标准 Young 表格的经典概念给出 G= SZ 的单独证明。当然，对于应用来说，经典概念更合适。在附录中，我们给出了 Seshadri 等人意义上的标准杨氏画面概念的“翻译”。代入 G= Sp2,, 和 Spin, 的杨氏画面的经典概念。在 3.8 中我们也给出了 G= G, 的这样一个翻译。对于其他特殊群体，作者并不知道这样的翻译。
For the general linear group GZ, the Littlewood-Richardson rule (see [18, 193) gives a method to calculate the decomposition of a tensor product of two irreducible G/,-representations. The aim of this article is to give a generalization of this rule for all simple, simply connected algebraic groups of type A,, B,, C,, D,, G2, and E,. We obtain also partial results for G of type F4, E,, and ES. The restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard Young tableau. Such a notion has been developed by Seshadri, Lakshmibai, Musili, and Rajeswari in a series of articles (see [ll, 13, 14, 16]), but not yet for all representations of the last three exceptional groups.The advantage of the notion of a Young tableau developed by Seshadri et al. is that it is independent of the type of the group. For the convenience of the reader not used to this notion we give first a seperate proof for G= SZ, using the classical notion of a standard Young tableau. Of course, for applications the classical notion is much more appropriate. In the Appendix we give a “translation” of the notion of a standard Young tableau in the sense of Seshadri et al. into the classical notion of a Young tableau for G= Sp2,,, and Spin,. In 3.8 we give also such a translation for G= G,. For the other exceptional groups such a translation is not known to the author.