Symmetry in Nonlinear Mathematical Physics - 2009
Maryna Nesterenko (Institute of Mathematics, Kyiv, Ukraine)
Jiri Patera (Centre de Recherches Mathématiques,
Université de Montréal, Canada)
Agnieszka Tereszkiewicz (Institute of Mathematics, University of Bialystok, Poland)
Orbit functions of SU(n) and Chebyshev polynomials
Three pairs of infinite families of special functions, based on the
Lie groups SU(n),
depending on n variables (n < ∞), are recalled.
Then, it is shown that the functions of each pair coincide one by one.
A substitution of variables transforms the functions into
n-dimensional Chebyshev polynomials.
One family of each pair is built, using orbits of the group of
permutations Sn of n objects
(or its alternating subgroup), while the second family of the pair is
built using orbits
of the Weyl group W(SU(n)) (or its even subgroup We). The
definition starting from Sn
leads naturally to variables relative to an orthonormal basis, while
the W(SU(n)) definition uses
the non-orthogonal basis of the simple roots of SU(n). The
correspondence is valid
when the functions of each pair depend on continuous variables, x Î Rn,
and also when the functions are sampled on lattices of matching
densities in Rn.
Discretization of Chebyshev polynomials is a direct consequence of
discretization of the orbit functions.
See for more details arXiv:0905.2925.