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.