Compact Simple Lie Groups and Their C-, S-, and E-Transforms

New continuous group transforms, together with their discretization over a lattice of any density and admissible symmetry, are defined for a general compact simple Lie groups of rank $2\leq n<\infty$. Rank 1 transforms are known. Rank 2 exposition of $C$- and $S$-transforms is in the literature. The $E$-transforms appear here for the first time.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Orbit Functions

In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space $E_n$ are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group $G$ of rank $n$ from one of its Weyl group orbits. It is shown that values of 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 $E_n$. Orbit functions are solutions of the corresponding Laplace equation in $E_n$, satisfying the Neumann condition on the boundary of $F$. Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Antisymmetric Orbit Functions

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

E-Orbit Functions

We review and further develop the theory of $E$-orbit functions. They are functions on the Euclidean space $E_n$ obtained from the multivariate exponential function by symmetrization by means of an even part $W_{e}$ of a Weyl group $W$, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group $W$. The $E$-orbit functions, determined by integral parameters, are invariant with respect to even part $W^{\rm aff}_{e}$ of the affine Weyl group corresponding to $W$. The $E$-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain $F^{e}$ of the group $W^{\rm aff}_{e}$ (the discrete $E$-orbit function transform).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group $G$. Such functions can always be restricted without loss of information to a fundamental region $\check F$ of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over $\check F$ (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in $\check F$ (discrete orthogonality), arising as the set of points in $\check F$ representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus $\mathbb T$. The characters of the centre $Z$ of the Lie group allow one to split functions $f$ on $\check F$ into a sum $f=f_1+...+f_c$, where $c$ is the order of $Z$, and where the component functions $f_k$ decompose into the series of $C$-, or $S$-, or $E$-functions from one congruence class only.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices

We consider the grading of $sl(n,\mathbb{C})$ by the group $\Pi_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $\Pi_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $\Pi_3$ and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in $\Pi_3$