|
SIGMA 2 (2006), 006, 60 pages math-ph/0601037
https://doi.org/10.3842/SIGMA.2006.006
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 January 04, 2006; Published online January 19, 2006
Abstract
In the paper, properties of orbit functions are reviewed
and further developed. Orbit functions on the Euclidean space
En are symmetrized exponential functions. The symmetrization is
fulfilled by a Weyl group corresponding to a Coxeter-Dynkin
diagram. Properties of such functions will be described. An orbit
function is the contribution to an irreducible character of a
compact semisimple Lie group G of rank n from one of its Weyl
group orbits. It is shown that values of 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. Orbit functions are solutions of the
corresponding Laplace equation in En, satisfying the Neumann
condition on the boundary of F. Orbit functions determine a
symmetrized Fourier transform and a transform on a finite set of
points.
Key words:
orbit functions; Coxeter-Dynkin diagram; Weyl group; orbits; products of orbits; orbit function transform; finite orbit function transform; Neumann boundary problem; symmetric polynomials.
pdf (599 kb)
ps (375 kb)
tex (56 kb)
References
- Patera J., Orbit functions of compact semisimple Lie groups
as special functions, in Proceedinds 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.
- Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and
special functions, Vols. 1-3, Dordrecht, Kluwer, 1991-1993.
- Miller W., Lie theory and special functions, New York, Academic
Press, 1968.
- Vilenkin N.Ja., Special functions and the theory of group
representations, Providence RI, Amer. Math. Soc., 1968.
- Macdonald I.G., Symmetric functions and Hall polynomials, 2nd
ed., Oxford, Oxford Univ. Press, 1995.
- Macdonald I.G., A new class of symmetric functions, Publ.
I.R.M.A. Strasbourg, 372/S-20, Actes 20, 1988, Séminaire
Lotharingien, 131-171.
- Macdonald I.G., Orthogonal polynomials associated with root
systems, Séminaire Lotharingien de Combinatoire, Actes
B45a, Strasbourg, 2000.
- Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and
special functions: recent advances, Dordrecht, Kluwer, 1995.
- Moody R.V., Patera J., Computation of character decompositions of
class functions on compact semisimple Lie groups, Math.
Comp., 1987, V.48, 799-827.
- Moody R.V., Patera J., Elements of finite order in Lie groups and
their applications, XIII Int. Colloq. on Group Theoretical Methods
in Physics, Editor W. Zachary, Singapore, World Scientific
Publishers, 1984, 308-318.
- 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
R.I., Amer. Math. Society, 1985, 227-264.
- McKay W.G., Moody R.V., Patera J., Decomposition of tensor
products of E8 representations, Algebras Groups Geom.,
1986, V.3, 286-328.
- Patera J., Sharp R.T., Branching rules for representations of
simple Lie algebras through Weyl group orbit reduction,
J. Phys. A: Math. Gen., 1989, V.22, 2329-2340.
- Grimm S., Patera J., Decomposition of tensor products of the
fundamental representations of E8, CRM Proc. Lecture
Notes, 1997, V.11, 329-355.
- 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.
- Rao K.R., Yip P., Discrete cosine transform - algorithms,
advantages, applications, New York, Academic Press, 1990.
- Kane R., Reflection groups and invariants, New York, Springer,
2002.
- Humphreys J.E., Reflection groups and Coxeter groups, Cambridge,
Cambridge Univ. Press, 1990.
- Humphreys J.E., Introduction to Lie algebras and representation
theory, New York, Springer, 1972.
- Pinsky M.A., The eigenvalues of an equilateral triangle, SIAM
J. Math. Anal., 1980, V. 11, 819-827.
- Patera J., Algebraic solution of the Neumann boundary problems on
fundamental regions of a compact semisimple Lie group, CRM
Preprint, Montreal, 2003.
- Patera J., Compact simple Lie groups and their C-, S-, and
E-transforms, SIGMA, 2005, V.1, paper 025, 6 pages,
math-ph/0512029.
- Bremner M.R., Moody R.V., Patera J., Tables of dominant weight
multiplicities for representations of simple Lie algebras, New York, Marcel
Dekker, 1985.
- McKay W.G., Patera J., Rand D.W., Tables of representations of
simple Lie algebras, Montreal, CRM, 1990.
- Champagne B., Kjiri M., Patera J., Sharp R.T., Description of
reflection generated polytopes using decorated Coxeter diagrams,
Can. J. Phys., 1995, V.73, 566-584.
- Moody R.V., Patera J., Voronoi and Delaunay cells of root
lattices: classification of their faces and facets by
Coxeter-Dynkin diagrams, J. Phys. A: Math. Gen., 1992,
V.25, 5089-5134.
- 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.
- Gingras F., Patera J., Sharp R.T., Orbit-orbit branching rules
between simple low-rank algebras and equal-rank subalgebras,
J. Math. Phys., 1992, V.33, 1618-1626.
- Patera J., Sharp R.T., Branching rules for representations of
simple Lie algebras through Weyl group orbits reduction, J.
Phys. A: Math. Gen., 1989, V.22, 2329-2340.
- Patera J., Zaratsyan A., Discrete and continuous cosine transform
generalized to Lie groups SU(2)×SU(2) and O(5),
J. Math. Phys., 2005, V.46, 053514, 25 pages.
- Patera J., Zaratsyan A., Discrete and continuous cosine transform
generalized to Lie groups SU(2) and G2, J. Math. Phys.,
2005, V.46, 113506, 17 pages.
- Lemire F.W., Patera J., Congruence number, a generalisation of
SU(3) triality, J. Math. Phys., 1980, V.21, 2026-2027.
- Zhelobenko D.P., Compact Lie groups and their representations,
Moscow, Nauka, 1970.
- McKay W.G., Patera J., Tables of dimensions, indices and branching
rules for representations of simple Lie algebras, New York, Marcel
Dekker, 1981.
- Strang G., The discrete cosine transform, SIAM Review, 1999,
V.41, 135-147.
- Kac V., Automorphisms of finite order of semisimple Lie algebras,
J. Funct. Anal. Appl., 1969, V.3, 252-255.
- 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.
- McKay W.G., Moody R.V., Patera J., Pianzola A., The 785 conjugacy
classes of rational elements of finite order in E8,
Contemp. Math., 1990, V.110, 79-123.
- Heckman G.J., Opdam E.M., Root systems and hypergeometric
functions. I, Compos. Math., 1987, V.64, 329-352.
- Heckman G.J., Root systems and hypergeometric functions. II,
Compos. Math., 1987, V.64, 353-373.
- Gasper G., Rahman M., Basic hypergeometric functions, Cambridge,
Cambridge Univ. Press, 1990.
|
|