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

SIGMA 3 (2007), 023, 83 pages      math-ph/0702040

Antisymmetric Orbit Functions

Anatoliy Klimyk a and Jiri Patera b
a) Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv 03143, Ukraine
b) Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received December 25, 2006; Published online February 12, 2007

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space En are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group G of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space En. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in En, vanishing on the boundary of the fundamental domain F. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group G. They also determine a transform on a finite set of points of F (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

Key words: antisymmetric orbit functions; signed orbits; products of orbits; orbit function transform; finite orbit function transform; finite Fourier transforms; finite cosine transforms; finite sine transforms; symmetric functions.

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  1. Klimyk A.U., Patera J., Orbit functions, SIGMA 2 (2006), 006, 60 pages, math-ph/0601037.
  2. 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.
  3. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, Oxford, 1995.
  4. 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.
  5. Macdonald I.G., Orthogonal polynomials associated with root systems, Séminaire Lotharingien de Combinatoire, Actes B45a, Stracbourg, 2000.
  6. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions: recent advances, Kluwer, Dordrecht, 1995.
  7. 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.
  8. 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.
  9. McKay W.G., Moody R.V., Patera J., Decomposition of tensor products of E8 representations, Algebras Groups Geom. 3 (1986), 286-328.
  10. 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.
  11. 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.
  12. Rao K. R., Yip P., Disrete cosine transform - algorithms, advantages, applications, Academic Press, New York, 1990.
  13. 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.
  14. 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, 17 pages.
  15. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to Lie groups SU(2) and G2, J. Math. Phys. 46 (2005), 113506, 25 pages.
  16. Atoyan A., Patera J., Continuous extension of the discrete cosine transform, and its applications to data processing, in Group Theory and Numerical Analysis, CRM Proc. Lecture Notes, Vol. 39, Amer. Math. Soc., Providence, RI, 2005, 1-15.
  17. Atoyan A., Patera J., Sahakian V., Akhperjanian A., Fourier transform method for imaging atmospheric Cherenkov telescopes, Astroparticle Phys. 23 (2005), 79-95, astro-ph/0409388.
  18. Patera J., Zaratsyan A., Zhu H.-M., New class of interpolation methods based on discretized Lie group transform, in SPIE Electronic Imaging, 2006, 6064A-06, S1.
  19. Germain M., Patera J., Zaratsyan A., Multiresolution analysis of digital images using the continuous extension of discrete group transform, in SPIE Electronic Imaging, 2006, 6065A-03, S2.
  20. Germain M., Patera J., Allard Y., Cosine transform generalized to Lie groups SU(2)×SU(2), O(5), and SU(2)×SU(2)×SU(2): application to digital image processing, Proc. SPIE 6065 (2006), 387-395.
  21. Moody R.V., Patera J., Computation of character decompositions of class functions on compact semisimple Lie groups, Math. Comp. 48 (1987), 799-827.
  22. 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.
  23. Kashuba I., Patera J., Discrete and continuous exponential transforms of simple Lie groups of rank two, math-ph/0702016.
  24. Kane R., Reflection groups and invariants, Springer, New York, 2002.
  25. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge University 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. Patera J., Compact simple Lie groups and their C-, S-, and E-transforms, SIGMA 1 (2005), 025, 6 pages, math-ph/0512029.
  30. Weyl H., The classical groups, Princeton University Press, 1939.
  31. 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.
  32. 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, Academic Press, New York, 1977, 235-278.
  33. Zhelobenko D.P., Compact Lie groups and their representations, Nauka, Moscow, 1970.
  34. Strang G., The discrete cosine transform, SIAM Rev. 41 (1999), 135-147.
  35. Martuchi S.A., Symmetric convolution and the discrete sine and cosine transforms, IEEE Trans. Signal Process. 42 (1994), 1038-1051.
  36. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions, Vol. 2, Kluwer, Dordrecht, 1993.
  37. Karlin S., McGregor J., Determinants of orthogonal polynomials, Bull. Amer. Math. Soc. 68 (1962), 204-209.
  38. Koornwinder T., Two-variable analogues of the classical orthogonal polynomials, in Theory and Applications of Special Functions, Editor R.A. Askey, Academic Press, New York, 1975, 435-495.
  39. Berens H., Schmid H., Xu Y., Multivariate Gaussian cubature formulas, Arch. Math. 64 (1995), 26-32.

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