Orbit functions of SU(n) and Chebyshev polynomials

Abstract

Orbit functions of a simple Lie group/Lie algebra L consist of exponential functions summed up over the Weyl group of L. They are labeled by the highest weights of irreducible finite dimensional representations of L. They are of three types: C-, S-and E-functions. Orbit functions of the Lie algebras A n , or equivalently, of the Lie group SU(n + 1), are considered. First, orbit functions in two different bases -one orthonormal, the other given by the simple roots of SU(n) -are written using the isomorphism of the permutation group of n elements and the Weyl group of SU(n). Secondly, it is demonstrated that there is a one-to-one correspondence between classical Chebyshev polynomials of the first and second kind, and C-and S-functions of the simple Lie group SU(2). It is then shown that the well-known orbit functions of SU(n) are straightforward generalizations of Chebyshev polynomials to n − 1 variables. Properties of the orbit functions provide a wealth of properties of the polynomials. Finally, multivariate exponential functions are considered, and their connection with orbit functions of SU(n) is established.

Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions

“…When functions of only two variables are considered, the structure of our special functions is quire transparent. For this reason, we avoided any reference to the underlying symmetric group S 3 of permutations of three elements [16].…”

“…A comparison of theoretical properties of the 'cosine transforms of S 3 ' in this paper and those of the standard S 2 × S 2 would be of interest and has yet to be made. The arguments in favor of the S 3 version of the transforms quote the ease of generalization to any dimension [6] and the possibility to work with data on lattices of other symmetries [16]. In general the greater the symmetry group underlying the formalism, the more economies one may expect in some applications.…”

“…The orbit functions of the paper have other useful properties that were not exposed here, two of which are: (i) Their products are decomposable into their finite sums. (ii) They can be naturally rewritten in terms of variables referring to non-orthogonal bases related to the simple roots of the simple Lie group SU (3), see [16].…”

“…So called E−functions [12] which are based on even subgroup of the Weyl group W are analogous to 'alternating' exponential functions [8,9]. In some cases, a direct relation between these two concepts can be found [16]. The coincidences of the orbit functions are due to the known isomorphism of the symmetric group S n and the Weyl group of the Lie group SU (n), and of the alternating subgroup of S n and the even subgroup of the Weyl group.…”

Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

“…It was shown in [19,20] that there is a one-to-one correspondence between C-and S-polynomials of A 1 and Chebyshev polynomials of the first and second kind respectively. Here we just write down the explicit form of the C-polynomials of A 1 (the Chebyshev polynomials of the first kind) traditionally denoted by T m , m = 0, 1, 2, .…”