36 research outputs found
Cubature formulas of multivariate polynomials arising from symmetric orbit functions
The paper develops applications of symmetric orbit functions, known from
irreducible representations of simple Lie groups, in numerical analysis. It is
shown that these functions have remarkable properties which yield to cubature
formulas, approximating a weighted integral of any function by a weighted
finite sum of function values, in connection with any simple Lie group. The
cubature formulas are specialized for simple Lie groups of rank two. An optimal
approximation of any function by multivariate polynomials arising from
symmetric orbit functions is discussed.Comment: 19 pages, 4 figure
Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2
The characters of irreducible finite dimensional representations of compact
simple Lie group G are invariant with respect to the action of the Weyl group
W(G) of G. The defining property of the new character-like functions ("hybrid
characters") is the fact that W(G) acts differently on the character term
corresponding to the long roots than on those corresponding to the short roots.
Therefore the hybrid characters are defined for the simple Lie groups with two
different lengths of their roots. Dominant weight multiplicities for the hybrid
characters are determined. The formulas for "hybrid dimensions" are also found
for all cases as the zero degree term in power expansion of the "hybrid
characters".Comment: 15 page
ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS
The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra C2
Cubature rules for unitary Jacobi ensembles
We present Chebyshev type cubature rules for the exact integration of
rational symmetric functions with poles on prescribed coordinate hyperplanes.
Here the integration is with respect to the densities of unitary Jacobi
ensembles stemming from the Haar measures of the orthogonal and the compact
symplectic Lie groups.Comment: 10 pages, 1 tabl
Special functions of Weyl groups and their continuous and discrete orthogonality
Cette thèse s'intéresse à l'étude des propriétés et applications de quatre familles des fonctions spéciales associées aux groupes de Weyl et dénotées , , et . Ces fonctions peuvent être vues comme des généralisations des polynômes de Tchebyshev. Elles sont en lien avec des polynômes orthogonaux à plusieurs variables associés aux algèbres de Lie simples, par exemple les polynômes de Jacobi et de Macdonald. Elles ont plusieurs propriétés remarquables, dont l'orthogonalité continue et discrète. En particulier, il est prouvé dans la présente thèse que les fonctions et caractérisées par certains paramètres sont mutuellement orthogonales par rapport à une mesure discrète. Leur orthogonalité discrète permet de déduire deux types de transformées discrètes analogues aux transformées de Fourier pour chaque algèbre de Lie simple avec racines des longueurs différentes. Comme les polynômes de Tchebyshev, ces quatre familles des fonctions ont des applications en analyse numérique. On obtient dans cette thèse quelques formules de >, pour des fonctions de plusieurs variables, en liaison avec les fonctions , et . On fournit également une description complète des transformées en cosinus discrètes de types V--VIII à dimensions en employant les fonctions spéciales associées aux algèbres de Lie simples et , appelées cosinus antisymétriques et symétriques. Enfin, on étudie quatre familles de polynômes orthogonaux à plusieurs variables, analogues aux polynômes de Tchebyshev, introduits en utilisant les cosinus (anti)symétriques.This thesis presents several properties and applications of four families of Weyl group orbit functions called -, -, - and -functions. These functions may be viewed as generalizations of the well-known Chebyshev polynomials. They are related to orthogonal polynomials associated with simple Lie algebras, e.g. the multivariate Jacobi and Macdonald polynomials. They have numerous remarkable properties such as continuous and discrete orthogonality. In particular, it is shown that the - and -functions characterized by certain parameters are mutually orthogonal with respect to a discrete measure. Their discrete orthogonality allows to deduce two types of Fourier-like discrete transforms for each simple Lie algebra with two different lengths of roots. Similarly to the Chebyshev polynomials, these four families of functions have applications in numerical integration. We obtain in this thesis various cubature formulas, for functions of several variables, arising from -, - and -functions. We also provide a~complete description of discrete multivariate cosine transforms of types V--VIII involving the Weyl group orbit functions arising from simple Lie algebras and , called antisymmetric and symmetric cosine functions. Furthermore, we study four families of multivariate Chebyshev-like orthogonal polynomials introduced via (anti)symmetric cosine functions
Cubature rules from Hall-Littlewood polynomials
Discrete orthogonality relations for Hall-Littlewood polynomials are
employed, so as to derive cubature rules for the integration of homogeneous
symmetric functions with respect to the density of the circular unitary
ensemble (which originates from the Haar measure on the special unitary group
). By passing to Macdonald's hyperoctahedral Hall-Littlewood
polynomials, we moreover find analogous cubature rules for the integration with
respect to the density of the circular quaternion ensemble (which originates in
turn from the Haar measure on the compact symplectic group ). The cubature formulas under consideration are exact for a
class of rational symmetric functions with simple poles supported on a
prescribed complex hyperplane arrangement. In the planar situations
(corresponding to and ), a determinantal
expression for the Christoffel weights enables us to write down compact
cubature rules for the integration over the equilateral triangle and the
isosceles right triangle, respectively.Comment: 30 pages, 7 table