Orbit functions of SU(n) and Chebyshev polynomials

TL;DR

This paper explores orbit functions of SU(n), demonstrating their relation to Chebyshev polynomials and extending properties of classical polynomials to multivariate cases within Lie group theory.

Contribution

It establishes a direct correspondence between Chebyshev polynomials and orbit functions of SU(2), and generalizes these functions to multiple variables for SU(n).

Findings

Orbit functions of SU(n) generalize Chebyshev polynomials to multiple variables.

A one-to-one correspondence between Chebyshev polynomials and SU(2) orbit functions is demonstrated.

Properties of orbit functions inform properties of multivariate 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 An, 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.