标题： 平面网络和可逆矩阵的多重Horn问题 显示英文标题

标题： Multiple Horn problems for planar networks and invertible matrices

摘要： 乘法多重Horn问题要求确定可逆矩阵三元组$A,B,C$的组合$AB, BC$和$ABC$的可能奇异值，这些矩阵具有给定的奇异值。 对于埃尔米特矩阵和的特征值存在类似的问题（加法问题），以及对于平面网络连接中多路径的最大权值（热带问题）。 对于平面网络多重Horn问题，我们建立了必要条件，并猜想对于足够大的网络，这些条件也是充分的。 这些条件由迹等式和菱形不等式（来自经典Horn问题的hive描述中熟知的）以及一组新的四面体等式给出。 此外，如果对平面网络的权重施加Gelfand-Zeitlin条件，四面体等式将转化为晶体理论中的八面体递推关系。 我们以带有势能的正品种的角度给出了结果的几何解释。 在这种方法中，菱形不等式来自于热带化势能的不等式$\Phi^t \leqslant 0$，而四面体等式则是某些普吕克关系的热带化结果。 对于乘法问题，我们引入了一个缩放参数$s$，并证明当$s$足够大时（对应于指数级大/小的奇异值），与乘法问题相关的Duistermaat-Heckman测度集中在八面体递推位置的小邻域内。 显示英文摘要

摘要： The multiplicative multiple Horn problem is asking to determine possible singular values of the combinations $AB, BC$ and $ABC$ for a triple of invertible matrices $A,B,C$ with given singular values. There are similar problems for eigenvalues of sums of Hermitian matrices (the additive problem), and for maximal weights of multi-paths in concatenations of planar networks (the tropical problem). For the planar network multiple Horn problem, we establish necessary conditions, and we conjecture that for large enough networks they are also sufficient. These conditions are given by the trace equalities and rhombus inequalities (familiar from the hive description of the classical Horn problem), and by the new set of tetrahedron equalities. Furthermore, if one imposes Gelfand-Zeitlin conditions on weights of planar networks, tetrahedron equalities turn into the octahedron recurrence from the theory of crystals. We give a geometric interpretation of our results in terms of positive varieties with potential. In this approach, rhombus inequalities follow from the inequality $\Phi^t \leqslant 0$ for the tropicalized potential, and tetrahedron equalities are obtained as tropicalization of certain Pl\"ucker relations. For the multiplicative problem, we introduce a scaling parameter $s$, and we show that for $s$ large enough (corresponding to exponentially large/small singular values) the Duistermaat-Heckman measure associated to the multiplicative problem concentrates in a small neighborhood of the octahedron recurrence locus.