The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices
复制标题
实对称和四元自对偶矩阵的喇叭问题
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
J. Zuber
中科院分区：
文献类型：
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作者：
R. Coquereaux;J. Zuber
Horn 问题，即在给定 A 和 B 的谱的情况下，重新研究两个矩阵的和 C " A`B 的特征值，比较实对称、复厄米特和自对偶四元数 3 ˆ 3 矩阵的情况。特别是，如果 A 和 B 独立且均匀地分布在其上，则 C 的特征值的概率分布函数 (PDF) 可以这样说。分别在正交群、酉群和辛群作用下的轨道？虽然可以通过使用相关轨道积分的显式公式来研究后两种情况（埃尔米特和四元数），但实数对称矩阵的情况也相当有趣，因为数值实验揭示了特征值的 PDF 发散的奇点的出现。无痕 3 ^ 3 矩阵的特征值可以根据代数函数（四次多项式的根）及其积分进行计算，并重现预期的奇异模式，我们还将该 PDF 与区域多项式的（重新缩放的）结构常数相关联。 Weyl SUpnq 角色。
Horn's problem, i.e., the study of the eigenvalues of the sum C " A`B of two matrices, given the spectrum of A and of B, is reexamined , comparing the case of real symmetric, complex Hermitian and self-dual quaternionic 3 ˆ 3 matrices. In particular, what can be said on the probability distribution function (PDF) of the eigenvalues of C if A and B are independently and uniformly distributed on their orbit under the action of, respectively, the orthogonal, unitary and symplectic group ? While the two latter cases (Hermitian and quaternionic) may be studied by use of explicit formulae for the relevant orbital integrals, the case of real symmetric matrices is much harder. It is also quite intriguing, since numerical experiments reveal the occurrence of singularities where the PDF of the eigenvalues diverges. Here we show that the computation of the PDF of the symmetric functions of the eigenvalues for traceless 3 ˆ 3 matrices may be carried out in terms of algebraic functions-roots of quartic polynomials-and their integrals. The computation is carried out in detail in a particular case, and reproduces the expected singular patterns. The divergences are of logarithmic or inverse power type. We also relate this PDF to the (rescaled) structure constants of zonal polynomials and introduce a zonal analogue of the Weyl SUpnq characters.
论霍恩问题及其体积函数
影响因子：
2.4
作者：
Coquereaux, Robert;McSwiggen, Colin;Zuber, Jean-Bernard
通讯作者：
Zuber, Jean-Bernard