arXiv 2101.044762021-01math.RTmath.GR

代数群的无重数表示

Multiplicity-free representations of algebraic groups

Martin W. Liebeck, Gary M. Seitz, Donna M. Testerman

备注：: 236 pages, to appear in Memoirs of the AMS

摘要：

设$K$是特征为零的代数闭域，$G$是$K$上的连通约化代数群.我们解决的问题，分类三元组$(G,H,V)$，其中$H$是一个适当的连通子群的$G$，和$V$是一个有限维的不可约$G$ -模块，这样的限制$V$ $H$是多重性自由-也就是说，它的每个组成因素出现的多重性1。大量的经典工作，可以追溯到外尔，Dynkin，豪，Stembridge和其他人，以及最近的工作的作者，可以设置在这方面。本文在$H$和$G$都是$A$型单代数群，且$H$不可约地嵌入$G$的情况下，确定了所有这样的三元组.虽然有许多有趣的三元组$(G,H,V)$，但定义嵌入$H<G$和$G<GL(V)$的表示的最高权重的可能性非常有限。例如，除了两个例外情况之外，两个权重只能支持至多两个基本权重;并且在许多示例中，权重中的一个或另一个对应于$G$或$H$的自然模的交替或对称平方。

英文摘要：

Let $K$ be an algebraically closed field of characteristic zero, and let $G$ be a connected reductive algebraic group over $K$. We address the problem of classifying triples $(G,H,V)$, where $H$ is a proper connected subgroup of $G$, and $V$ is a finite-dimensional irreducible $G$-module such that the restriction of $V$ to $H$ is multiplicity-free -- that is, each of its composition factors appears with multiplicity 1. A great deal of classical work, going back to Weyl, Dynkin, Howe, Stembridge and others, and also more recent work of the authors, can be set in this context. In this paper we determine all such triples in the case where $H$ and $G$ are both simple algebraic groups of type $A$, and $H$ is embedded irreducibly in $G$. While there are a number of interesting familes of such triples $(G,H,V)$, the possibilities for the highest weights of the representations defining the embeddings $H<G$ and $G<GL(V)$ are very restricted. For example, apart from two exceptional cases, both weights can only have support on at most two fundamental weights; and in many of the examples, one or other of the weights corresponds to the alternating or symmetric square of the natural module for either $G$ or $H$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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