Antisymmetric Orbit Functions

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.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (6)

• E-Orbit Functions
• Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group
• Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization
• Fourier transform method for imaging atmospheric Cherenkov telescopes
• Compact Simple Lie Groups and Their C-, S-, and E-Transforms
• Discrete and continuous exponential transforms of simple Lie groups of rank two

Cited by in corpus (20)

• E-Orbit Functions
• Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group
• Gaussian cubature arising from hybrid characters of simple Lie groups
• (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms
• Cubature formulas of multivariate polynomials arising from symmetric orbit functions
• Orthogonal polynomials of compact simple Lie groups: Branching rules for polynomials
• Generalized discrete orbit function transforms of affine Weyl groups
• Weight-Lattice Discretization of Weyl-Orbit Functions
• Discrete transforms and orthogonal polynomials of (anti)symmetric multivariate cosine functions
• On E-Discretization of Tori of Compact Simple Lie Groups
• On E-Discretization of Tori of Compact Simple Lie Groups: II
• Discrete cosine and sine transforms generalized to honeycomb lattice
• Orthogonal polynomials of compact simple Lie groups
• Four families of orthogonal polynomials of C2 and symmetric and antisymmetric generalizations of sine and cosine functions
• Four families of Weyl group orbit functions of B_3 and C_3
• Discrete and continuous exponential transforms of simple Lie groups of rank two
• Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
• Spectral measures associated to rank two Lie groups and finite subgroups of $GL(2,\mathbb{Z})$
• On immanant functions related to Weyl groups of $A_n$
• Complete Decompositions of Coxeter groups orbit products of $A_2,C_2,G_2$ and $H_2$ and some limits for them