The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices

Robert Coquereaux ; Jean-Bernard Zuber

Symmetry, integrability and geometry: methods and applications, Tome 15 (2019)

Résumé

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 $3\times 3$ 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 $3\times 3$ 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 $\mathrm{SU}(n)$ characters.

Détail

Keywords: Horn problem; honeycombs; polytopes; zonal polynomials; Littlewood–Richardson coefficients.

Bibliographie

[1] Baratta W., SpecialFunctions.nb, The University of Melbourne, 2008 https://researchers.ms.unimelb.edu.au/w̃baratta/

[2] Bauer M., Zuber J.-B., On products of delta distributions and resultants, in preparation

[3] Brézin E., Hikami S., “An extension of the Harish-Chandra–Itzykson–Zuber integral”, Comm. Math. Phys., 235 (2003), 125–137, arXiv: math-ph/0208002 | DOI | MR

[4] Bufetov A., Gorin V., Fourier transform on high-dimensional unitary groups with applications to random tilings, arXiv: 1712.09925

[5] Constantine A. G., “Some non-central distribution problems in multivariate analysis”, Ann. Math. Statist., 34 (1963), 1270–1285 | DOI | MR | Zbl

[6] Coquereaux R., Mathematica package “SymPol\$Package” http://www.cpt.univ-mrs.fr/c̃oque/Computer_programs/index.html

[7] Coquereaux R., McSwiggen C., Zuber J.-B., On {H}orn's problem and its volume function, arXiv: 1904.00752

[8] Coquereaux R., Zuber J.-B., “On some properties of $\rm SU(3)$ fusion coefficients”, Nuclear Phys. B, 912 (2016), 119–150, arXiv: 1605.05864 | DOI | MR | Zbl

[9] Coquereaux R., Zuber J.-B., “From orbital measures to Littlewood–{R}ichardson coefficients and hive polytopes”, Ann. Inst. Henri Poincaré D, 5 (2018), 339–386, arXiv: 1706.02793 | DOI | MR | Zbl

[10] Demmel J., Koev P., “Accurate and efficient evaluation of Schur and Jack functions”, Math. Comp., 75 (2006), 223–239 | DOI | MR | Zbl

[11] Dumitriu I., Edelman A., Shuman G., “MOPS: multivariate orthogonal polynomials (symbolically)”, J. Symbolic Comput., 42 (2007), 587–620, arXiv: math-ph/0409066 | DOI | MR | Zbl

[12] Faraut J., “Horn's problem and Fourier analysis”, Tunis. J. Math., 1 (2019), 585–606 | DOI | MR | Zbl

[13] Féray V., Śniady P., “Zonal polynomials via {S}tanley's coordinates and free cumulants”, J. Algebra, 334 (2011), 338–373, arXiv: 1005.0316 | DOI | MR

[14] Frumkin A., Goldberger A., Diagonals of real symmetric matrices of given spectra as a measure space, arXiv: 1505.06418

[15] Frumkin A., Goldberger A., “On the distribution of the spectrum of the sum of two Hermitian or real symmetric matrices”, Adv. in Appl. Math., 37 (2006), 268–286 | DOI | MR | Zbl

[16] Fulton W., “Eigenvalues, invariant factors, highest weights, and Schubert calculus”, Bull. Amer. Math. Soc., 37 (2000), 209–249, arXiv: math.AG/9908012 | DOI | MR | Zbl

[17] Gel'fand I. M., Shilov G. E., Generalized functions, v. I, Properties and operations, Academic Press, New York–London, 1964 | MR

[18] Gorin V., Marcus A. W., Crystallization of random matrix orbits, arXiv: 1706.07393

[19] Harish-Chandra, “Differential operators on a semisimple Lie algebra”, Amer. J. Math., 79 (1957), 87–120 | DOI | MR | Zbl

[20] Hikami S., Brézin E., “WKB-expansion of the Harish-Chandra–Itzykson–Zuber integral for arbitrary $\beta$”, Progr. Theoret. Phys., 116 (2006), 441–502, arXiv: math-ph/0604041 | DOI | MR | Zbl

[21] Hua L.-K., Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc., Providence, R.I., 1963 | MR

[22] Itzykson C., Zuber J.-B., “The planar approximation. II”, J. Math. Phys., 21 (1980), 411–421 | DOI | MR | Zbl

[23] James A. T., “Normal multivariate analysis and the orthogonal group”, Ann. Math. Statist., 25 (1954), 40–75 | DOI | MR | Zbl

[24] James A. T., “The distribution of the latent roots of the covariance matrix”, Ann. Math. Statist., 31 (1960), 151–158 | DOI | MR | Zbl

[25] James A. T., “Zonal polynomials of the real positive definite symmetric matrices”, Ann. of Math., 74 (1961), 456–469 | DOI | MR | Zbl

[26] Klyachko A. A., “Stable bundles, representation theory and Hermitian operators”, Selecta Math. (N.S.), 4 (1998), 419–445 | DOI | MR | Zbl

[27] Knutson A., Tao T., “The honeycomb model of ${\rm GL}_n({\mathbb C})$ tensor products. I Proof of the saturation conjecture”, J. Amer. Math. Soc., 12 (1999), 1055–1090, arXiv: math.RT/9807160 | DOI | MR | Zbl

[28] Knutson A., Tao T., “Honeycombs and sums of Hermitian matrices”, Notices Amer. Math. Soc., 48 (2001), 175–186, arXiv: math.RT/0009048 | MR | Zbl

[29] Knutson A., Tao T., Woodward C., “The honeycomb model of ${\rm GL}_n({\mathbb C})$ tensor products. II Puzzles determine facets of the Littlewood–Richardson cone”, J. Amer. Math. Soc., 17 (2004), 19–48, arXiv: math.CO/0107011 | DOI | MR | Zbl

[30] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979 | MR

[31] Mathai A. M., Provost S. B., Hayakawa T., Bilinear forms and zonal polynomials, Lecture Notes in Statistics, 102, Springer-Verlag, New York, 1995 | DOI | MR | Zbl

[32] Okounkov A., Olshanski G., “Shifted Jack polynomials, binomial formula, and applications”, Math. Res. Lett., 4 (1997), 69–78, arXiv: q-alg/9608020 | DOI | MR | Zbl

[33] Stanley R. P., “Some combinatorial properties of Jack symmetric functions”, Adv. Math., 77 (1989), 76–115 | DOI | MR | Zbl

[34] Zuber J.-B., “Horn's problem and Harish-Chandra's integrals. Probability density functions”, Ann. Inst. Henri Poincaré D, 5 (2018), 309–338, arXiv: 1705.01186 | DOI | MR | Zbl