Four types of special functions ofG₂and their discretization

Abstract

Abstract.Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G 2 , are compared and described. Two of the four families (called here C-and S-functions) are well known. New results of the paper are in description of two new families of G 2 -functions not found in the literature. They are denoted as S L -and S S -functions.It is shown that all four families have analogous useful properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space. They are also orthogonal as discrete functions when their values are sampled at the lattice points F M ⊂ F and added up with appropriate weight function. The weight functions are determined for the new families. Products of ten types among the four families of functions, namelyand S L S L , are completely decomposable into the finite sum of the functions belonging to just one of the families. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.

Multidimensional Hybrid Boundary Value Problem

“…They are symmetric and skew-symmetric with respect to the finite reflection group [6,10,16,[19][20][21]23]. The other two, S s -and S l -functions [11,12,16,17,23] have analogous properties as C-and S-functions. The main difference between them is their behaviour at the boundary of their domain of orthogonality in R n .…”

“…There are four kinds of special functions of interest to us whose orthogonality on lattice fragment F is known for any simple Lie group [5,6,10,16,17,19,20,23, and references therein]. The general formula for special functions (called orbit functions) [10,11] corresponding to the finite reflection group G is given by w∈G σ(w)e 2πi wλ|x , λ ∈ P + , x ∈ F…”

“…The solutions are presented as expansions into a series of special functions that satisfy required conditions at the (n − 1)-dimensional boundaries of F . The recent discovery of special functions [5,10,11,16,19,23] makes realization of this idea easy and straightforward in any dimension. The new functions, called 'multidimensional hybrids', satisfy the Dirichlet boundary condition on some parts of the boundary F and Neumann on the remaining ones.…”

Dominant Weight Multiplicities in Hybrid Characters of B n , C n , F 4, G 2

“…In [23] the hybrid characters are described for all simple Lie algebras with 2-root lengths. The definitions use the homomorphisms (1) denoted by σ L and σ S , Also the G 2 case is found in [31]. For every representation of those groups there are two new character-like functions, χ L λ (g) and χ S λ (g), in addition to their corresponding character χ λ (g) functions.…”

Discrete Fourier Analysis and Chebyshev Polynomials with G2Group

“…The results include cubature rules and orthogonal trigonometric functions that are analogues of cosine and sine functions. There are four families of such functions and they have also been studied recently in [13,18]. While the results in these two papers concern mainly with orthogonal polynomials, our emphasis is on the discrete Fourier analysis and cubature rules, and on the connection to the results in the hexagonal domain.…”