Antisymmetric Orbit Functions

TL;DR

This paper reviews and develops properties of antisymmetric orbit functions, which are related to Lie group characters, solutions to Laplace's equation, and Fourier transforms, with applications to symmetric and antisymmetric discrete transforms.

Contribution

It provides a comprehensive analysis of antisymmetric orbit functions, linking them to Lie group characters, differential equations, and Fourier analysis, including new discrete transform methods.

Findings

Antisymmetric orbit functions are solutions to Laplace's equation vanishing on fundamental domain boundaries.

They are related to irreducible characters of compact semisimple Lie groups.

The paper introduces symmetric and antisymmetric multivariate discrete transforms.

Abstract

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ 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 $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, 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.