The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices
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 re-examined, comparing the case of real symmetric, complex Hermitian, and self-dual quaternionic 3×3 matrices. In particular, what can be said on the probability distributio...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210193 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices / R. Coquereaux, J.-B. Zuber // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 34 назв. — англ. |
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Coquereaux, R. Zuber, J.-B. 2025-12-03T14:34:25Z 2019 The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices / R. Coquereaux, J.-B. Zuber // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B08; 17B10; 22E46; 43A75; 52Bxx arXiv: 1809.03394 https://nasplib.isofts.kiev.ua/handle/123456789/210193 https://doi.org/10.3842/SIGMA.2019.029 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 re-examined, comparing the case of real symmetric, complex Hermitian, and self-dual quaternionic 3×3 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 groups? While the latter two 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 3×3 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 SU(n) characters. Many thanks to Michèle Vergne for challenging us to carry out the n = 3 calculation and for her patient explanations on the location of the singularities occurring in orbital integrals, and to Michel Bauer for his encouragement and, in particular, for his constructive criticism of identity (4.9). Stimulating conversations with O. Babelon, É. Brézin, P. Di Francesco, J. Faraut, V. Gorin, S. Majumdar, and G. Schehr are also acknowledged. We thank the referees for suggesting several editorial improvements. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices |
| spellingShingle |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices Coquereaux, R. Zuber, J.-B. |
| title_short |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices |
| title_full |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices |
| title_fullStr |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices |
| title_full_unstemmed |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices |
| title_sort |
horn problem for real symmetric and quaternionic self-dual matrices |
| author |
Coquereaux, R. Zuber, J.-B. |
| author_facet |
Coquereaux, R. Zuber, J.-B. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 re-examined, comparing the case of real symmetric, complex Hermitian, and self-dual quaternionic 3×3 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 groups? While the latter two 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 3×3 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 SU(n) characters.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210193 |
| citation_txt |
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices / R. Coquereaux, J.-B. Zuber // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 34 назв. — англ. |
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2025-12-07T21:24:42Z |
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