Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 085, 12 pages      math.CA/0606391      http://dx.doi.org/10.3842/SIGMA.2006.085
Contribution to the Vadim Kuznetsov Memorial Issue

Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices

Hjalmar Rosengren
Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden

Received October 11, 2006; Published online December 04, 2006

Abstract
We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares).

Key words: Christoffel-Darboux kernel; multivariable orthogonal polynomial; Pfaffian; determinant; correlation function; random Hermitian matrix; orthogonal polynomial ensemble; Sundquist's identities.

pdf (241 kb)   ps (167 kb)   tex (14 kb)

References

  1. Baik J., Deift O., Strahov E., Products and ratios of characteristic polynomials of random Hermitian matrices, J. Math. Phys., 2003, V.44, 3657-3670, math-ph/0304016.
  2. Borodin A., Strahov E., Averages of characteristic polynomials in random matrix theory, Comm. Pure Appl. Math., 2006, V.59, 161-253, math-ph/0407065.
  3. Bourbaki N., Éléments de mathématique, fascicule VII, livre II, chapitre III: Algèbre multilinéaire, Paris, Hermann, 1958.
  4. Brézin E., Hikami S., Characteristic polynomials of random matrices, Comm. Math. Phys., 2000, V.214, 111-135, math-ph/9910005.
  5. Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, New York University, 1999.
  6. Ishikawa M., Okada S., Tagawa H., Zeng J., Generalizations of Cauchy's determinant and Schur's Pfaffian, Adv. Appl. Math., 2006, V.36 251-287, math.CO/0411280.
  7. Ishikawa M., Wakayama M., Minor summation formulas of Pfaffians, survey and a new identity, in Combinatorial Methods in Representation Theory, Editors K. Koike et al., Adv. Stud. Pure Math., Vol. 28, Kinokuniya, Tokyo, 2000, 133-142.
  8. Ishikawa M., Wakayama M., Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities, J. Combin. Theory Ser. A, 2006, V.13, 113-155, math.CO/0312358.
  9. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Cambridge, Cambridge University Press, 2005.
  10. Izergin A.G., Partition function of a six-vertex model in a finite volume, Soviet Phys. Dokl., 1987, V.32, 878-879.
  11. Kac V.G., Wakimoto M., Integrable highest weight modules over affine superalgebras and number theory, in Lie Theory and Geometry, Editors J.-L. Brylinski et al., Progr. Math., Vol. 123, Boston, MA, Birkhäuser, 1994, 415-456.
  12. König W., Orthogonal polynomial ensembles in probability theory, Probab. Surv., 2005, V.2, 385-447, math.PR/0403090.
  13. Kuperberg G., Another proof of the alternating-sign matrix conjecture, Internat. Math. Res. Notices, 1996, V.1996, 139-150, math.CO/9712207.
  14. Kuperberg G., Symmetry classes of alternating-sign matrices under one roof, Ann. Math., 2002, V.156, 835-866, math.CO/0008184.
  15. Lascoux A., Symmetric functions and combinatorial operators on polynomials, Providence, American Mathematical Society, 2003.
  16. Lascoux A., Pfaffians and representations of the symmetric group, math.CO/0610510.
  17. Lascoux A., He S., Généralisation de la formule de Darboux-Christoffel pour les polynômes orthogonaux, C. R. Acad. Sci. Paris Sér. I Math., 1985, V.300, 681-683.
  18. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford, Oxford University Press, 1995.
  19. Milne S.C., New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function, Proc. Nat. Acad. Sci. U.S.A., 1996, V.93, 15004-15008.
  20. Milne S.C., New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function, in Formal Power Series and Algebraic Combinatorics, 9th Conference, Vol. 3, Universität Wien, 1997, 403-417.
  21. Milne S.C., Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Ramanujan J., 2002, V.6, 7-149, math.NT/0008068.
  22. Okada S., Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra, 1998, V.205, 337-367.
  23. Okada S., Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, J. Algebraic Combin., 2006, V.23, 43-69, math.CO/0408234.
  24. Rains E., Quadratic Pfaffian identities, unpublished manuscript.
  25. Rosengren H., Sums of triangular numbers from the Frobenius determinant, Adv. Math., 2007, V.208, 935-961, math.NT/0504272.
  26. Rosengren H., Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux, math.CO/0603086.
  27. Rosengren H., Sums of squares from elliptic pfaffians, math.NT/0610278.
  28. Stembridge J.R., Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math., 1990, V.83, 96-131.
  29. Strahov E., Fyodorov Y.V., Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach, Comm. Math. Phys., 2003, V.241, 343-382, math-ph/0210010.
  30. Sundquist T., Two variable Pfaffian identities and symmetric functions, J. Algebraic Combin., 1996, V.5, 135-148.
  31. Zagier D., A proof of the Kac-Wakimoto affine denominator formula for the strange series, Math. Res. Lett., 2000, V.7, 597-604.
  32. Zeilberger D., Proof of the refined alternating sign matrix conjecture, New York J. Math., 1996, V.2, 59-68, math.CO/9606224.

Previous article   Next article   Contents of Volume 2 (2006)