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

SIGMA 2 (2006), 076, 14 pages      math-ph/0611020      https://doi.org/10.3842/SIGMA.2006.076

### Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

Robert V. Moody a and Jiri Patera b
a) Department of Mathematics, University of Victoria, Victoria, British Columbia, Canada
b) Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received October 30, 2006; Published online November 08, 2006

Abstract
The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group G. Such functions can always be restricted without loss of information to a fundamental region F of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over F (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in F (discrete orthogonality), arising as the set of points in F representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus T. The characters of the centre Z of the Lie group allow one to split functions f on F into a sum f = f1 + ¼ + fc, where c is the order of Z, and where the component functions fk decompose into the series of C-, or S-, or E-functions from one congruence class only.

Key words: orbit functions; Weyl group; semisimple Lie group; continuous orthogonality; discrete orthogonality.

pdf (260 kb)   ps (187 kb)   tex (17 kb)

References

1. Patera J., Compact simple Lie groups and their C-, S-, and E-transforms, SIGMA, 2005, V.1, Paper 025, 6 pages, math-ph/0512029.
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, 2004, V.50, Part 3, 1152-1160.
3. Moody R.V., Patera J., Computation of character decompositions of class functions on compact semisimple Lie groups, Math. Comp., 1987, V.48, 799-827.
4. Moody R.V., Patera J., Characters of elements of finite order in simple Lie groups, SIAM J. Algebraic Discrete Methods, 1984, V.5, 359-383.
5. Moody R.V., Patera J., Elements of finite order in Lie groups and their applications, in Proceedings XIII International Colloquium on Group Theoretical Methods in Physics (College Park, 1984), Editor W. Zachary, Singapore, World Scientific Publishers, 1984, 308-318.
6. Kass S., Moody R.V., Patera J., Slansky R., Affine Lie algebras, weight multiplicities, and branching rules, Vol. I and II, Los Alamos Series in Basic and Applied Sciences, Berkeley, University of California Press, 1990.
7. 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, Providence, RI, Amer. Math. Society, 1985, 227-264.
8. 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, Providence, RI, Amer. Math. Soc., 1997, 329-355.
9. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to the Lie groups SU(3) and G(2), J. Math. Phys., 2005, V.46, 113506, 17 pages.
10. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to the Lie groups SU(2)×SU(2) and O(5), J. Math. Phys., 2005, V.46, 053514, 25 pages.
11. Patera J., Zaratsyan A., Discrete and continuous sine transform generalized to semisimple Lie groups of rank two, J. Math. Phys., 2006, V.47, 043512, 22 pages.
12. Kashuba I., Patera J., Discrete and continuous E-transforms of semisimple Lie group of rank two, to appear.
13. Klimyk A., Patera J., Orbit functions, SIGMA, 2006, V.2, Paper 006, 60 pages, math-ph/0601037.
14. Klimyk A., Patera J., Antisymmetric orbit functions, SIGMA, 2007, V.3, to appear.
15. 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, Providence, RI, Amer. Math. Soc., 2005, 1-15.
16. Atoyan A., Patera J., Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization, J. Math. Phys., 2004, V.45, 2468-2491, math-ph/0309039.
17. Atoyan A., Patera J., Sahakian V., Akhperjanian A., Fourier transform method for imaging atmospheric Cherenkov telescopes, Astroparticle Phys., 2005, V.23, 79-95, astro-ph/0409388.
18. Patera J., Zaratsyan A., Zhu H.-M., New class of interpolation methods based on discretized Lie group transforms, in SPIE Electronic Imaging (2006, San Jose), 2006, 6064A-06, S1.
19. Germain M., Patera J., Zaratsyan A., Multiresolution analysis of digital images using the continuous extension of discrete group transforms, in SPIE Electronic Imaging (2006, San Jose), 2006, 6065-03, 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, 2006, V.6065, 387-395.
21. Humphreys J., Introduction to Lie algebras and representation theory, New York, Springer, 1972.
22. Bremner M., Moody R.V., Patera J., Dominant weight multiplicities, New York, Marcel Dekker, 1990.
23. Moody R.V., Patera J., Group theory of multidimensional fast Fourier transforms, in preparation.