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

SIGMA 4 (2008), 002, 57 pages      arXiv:0801.0822      https://doi.org/10.3842/SIGMA.2008.002

E-Orbit Functions

Anatoliy U. Klimyk a and Jiri Patera b
a) Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv 03680, Ukraine
b) Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received December 20, 2007; Published online January 05, 2008

Abstract
We review and further develop the theory of E-orbit functions. They are functions on the Euclidean space En obtained from the multivariate exponential function by symmetrization by means of an even part We of a Weyl group W, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group W. The E-orbit functions, determined by integral parameters, are invariant with respect to even part Weaff of the affine Weyl group corresponding to W. The E-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain Fe of the group Weaff (the discrete E-orbit function transform).

Key words: E-orbit functions; orbits; products of orbits; symmetric orbit functions; E-orbit function transform; finite E-orbit function transform; finite Fourier transforms.

pdf (572 kb)   ps (341 kb)   tex (52 kb)

References

  1. Patera J., Orbit functions of compact semisimple Lie groups as special functions, in Proceedings of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv 50 (2004), Part 3, 1152-1160.
  2. Patera J., Compact simple Lie groups and their C-, S-, and E-transforms, SIGMA 1 (2005), 025, 6 pages, math-ph/0512029.
  3. Atoyan A., Patera J., Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization, J. Math. Phys. 45 (2004), 2468-2491, math-ph/0309039.
  4. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to Lie groups SU(2) × SU(2) and O(5), J. Math. Phys. 46 (2005), 053514, 25 pages.
  5. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to Lie groups SU(2) and G2, J. Math. Phys. 46 (2005), 113506, 17 pages.
  6. Patera J., Zaratsyan A., Discrete and continuous sine transforms generalized to compact semisimple Lie groups of rank two, J. Math. Phys. 47 (2006), 043512, 22 pages.
  7. Klimyk A.U., Patera J., Orbit functions, SIGMA 2 (2006), 006, 60 pages, math-ph/0601037.
  8. Klimyk A.U., Patera J., Antisymmetric orbit functions, SIGMA 3 (2007), 023, 83 pages, math-ph/0702040.
  9. Kashuba I., Patera J., Discrete and continuous exponential transforms of simple Lie groups of rank two, J. Phys. A: Math. Theor. 40 (2007), 1751-1774, math-ph/0702016.
  10. Moody R.V., Patera J., Orthogonality within the families of C-, S-, and E-functions of any compact semisimple Lie group, SIGMA 2 (2006), 076, 14 pages, math-ph/0611020.
  11. Moody R.V., Patera J., Computation of character decompositions of class functions on compact semisimple Lie groups, Math. Comp. 48 (1987), 799-827.
  12. Klimyk A.U., Patera J., (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms, J. Phys. A: Math. Theor. 40 (2007), 10473-10489, arXiv:0705.3572.
  13. Klimyk A.U., Patera J., (Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms, J. Math. Phys. 48 (2007), 093504, 24 pages, arXiv:0705.4186.
  14. Macdonald I.G., Symmetric functions and hall polynomials, 2nd ed., Oxford Univ. Press, Oxford, 1995.
  15. Macdonald I.G., A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien, 1988, 131-171.
  16. Macdonald I.G., Orthogonal polynomials associated with root systems, Séminaire Lotharingien de Combinatoire, Actes B45a, Stracbourg, 2000.
  17. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions: recent advances, Kluwer, Dordrecht, 1995.
  18. Moody R.V., Patera J., Elements of finite order in Lie groups and their applications, in Proceedings of XIII Int. Colloq. on Group Theoretical Methods in Physics, Editor W. Zachary, World Scientific Publishers, Singapore, 1984, 308-318.
  19. McKay W.G., Moody R.V., Patera J., Tables of E8 characters and decomposition of plethysms, in Lie Algebras and Related Topics, Editors D.J. Britten, F.W. Lemire and R.V. Moody, Amer. Math. Society, Providence RI, 1985, 227-264.
  20. McKay W.G., Moody R.V., Patera J., Decomposition of tensor products of E8 representations, Algebras Groups Geom. 3 (1986), 286-328.
  21. Patera J., Sharp R.T., Branching rules for representations of simple Lie algebras through Weyl group orbit reduction, J. Phys. A: Math. Gen. 22 (1989), 2329-2340.
  22. Grimm S., Patera J., Decomposition of tensor products of the fundamental representations of E8, in Advances in Mathematical Sciences - CRM's 25 Years, Editor L. Vinet, CRM Proc. Lecture Notes, Vol. 11, Amer. Math. Soc., Providence, RI, 1997, 329-355.
  23. Rao K.R., Yip P., Discrete cosine transform - algorithms, advantages, applications, Academic Press, New York, 1990.
  24. Kane R., Reflection groups and invariants, Springer, New York, 2002.
  25. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Univ. Press, Cambridge, 1990.
  26. Humphreys J.E., Introduction to Lie algebras and representation theory, Springer, New York, 1972.
  27. Bremner M.R., Moody R.V., Patera J., Tables of dominant weight multiplicities for representations of simple Lie algebras, Marcel Dekker, New York, 1985.
  28. Kac V., Infinite dimensional Lie algebras, Birkhäuser, Basel, 1982.
  29. Mckay W.G., Patera J., Sannikoff D., The computation of branching rules for representations of semisimple Lie algebras, in Computers in Nonassociative Rings and Algebras, Editors R.E. Beck and B. Kolman, New York, Academic Press, 1977, 235-278.
  30. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions, Vol. 2, Kluwer, Dordrecht, 1993.

Previous article   Next article   Contents of Volume 4 (2008)