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SIGMA 3 (2007), 023, 83 pages math-ph/0702040
https://doi.org/10.3842/SIGMA.2007.023
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
Abstract
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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