The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G = SL(n). There is a polyhedral cone (the \eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G.We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face. The remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [B3] which handled the case of G = SL(n).

中文翻译：

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任意类型的埃尔米特特征值问题中的极值射线

埃尔米特特征值问题要求给出给定和的特征值的情况下，埃尔米特矩阵总和的可能特征值。这是关于G = SL（n）的最大紧致子群的李代数的问题。有一个多面体圆锥（\ eigencone”）来确定问题的可能答案。可以为任意半简单群G定义这些本征圆锥，并控制（张量的）张量积中存在非零不变变量的（适当稳定的）问题。G的不可约表示。我们给出任意半简单群G的本征圆锥的极射线的描述首先观察极端光线位于规则的小面上，然后对任意规则的面孔分类极端光线。给出了一些极端射线的显式，这些射线在任意规则的面上都具有显着的几何意义，作为有趣位点的循环类。我们引入的几何过程可以理解该面上剩余的极端射线，并在数值上进行解释，称为李维亚群的归纳法。给出了几个数值示例。本文的主要结果和方法概括了[B3]，它处理了G = SL（n）的情况。