A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
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DOI：
10.1007/bf01231564
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发表时间：
1994-12
影响因子：
3.1
通讯作者：
P. Littelmann
中科院分区：
文献类型：
--
作者：
P. Littelmann
令 G 为复可对称 Kac-Moody 代数。在这篇文章中，我们证明了一个Littlewood-Richardson型规则来计算G的两个简单、可积、最高权模的张量积分解为不可约分量。在群GLn(C)的表示论中，一个重要的工具是Young tableaux。不可简化的表征与这些画面的形状一一对应。令 T 为 GLn (C) 中对角矩阵的子群。然后有一种规范的方法将 T 的权重分配给任何 Young 画面，使得固定形状的所有画面的权重之和是相应 GLn (C)-模块 V 的字符 CharV。请注意，这不仅给出了计算字符的方法，还提供了描述表示中权重重数的可能性：它是相同权重的不同画面的数量。最终，Littlewood-Richardson 规则纯粹根据这些 Young 画面的组合来描述 GLn (C)-模张量积的分解。
Let G be a complex symmetrizable Kac-Moody algebra. In this article we prove a Littlewood-Richardson type rule to calculate the decomposition of the tensor product of two simple, integrable, highest weight modules of G into irreducible components.In the representation theory of the group GLn (C), an important tool are the Young tableaux. The irreducible representations are in one-to-one correspondence with the shapes of these tableaux. Let T be the subgroup of diagonal matrices in GLn (C). Then there is a canonical way to assign a weight of T to any Young tableau such that the sum over the weights of all tableaux of a fixed shape is the character CharV of the corresponding GLn (C)-module V. Note that this gives not only a way to compute the character, it gives also a possibility to describe the multiplicity of a weight in the representation: It is the number of different tableaux of the same weight. Eventually, the Littlewood-Richardson rule describes the decomposition of tensor products of GLn (C)-modules purely in terms of the combinatoric of these Young tableaux.