38 research outputs found
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 . Rank 1 transforms are
known. Rank 2 exposition of - and -transforms is in the literature. The
-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 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 of rank from one of its Weyl group
orbits. It is shown that values of orbit functions are repeated on copies of
the fundamental domain of the affine Weyl group (determined by the initial
Weyl group) in the entire Euclidean space . Orbit functions are solutions
of the corresponding Laplace equation in , satisfying the Neumann
condition on the boundary of . 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
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 of rank . Up to a sign,
values of antisymmetric orbit functions are repeated on copies of the
fundamental domain of the affine Weyl group (determined by the initial Weyl
group) in the entire Euclidean space . Antisymmetric orbit functions are
solutions of the corresponding Laplace equation in , vanishing on the
boundary of the fundamental domain . 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 . They also determine a transform on a finite set of points of
(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 -orbit functions. They are
functions on the Euclidean space obtained from the multivariate
exponential function by symmetrization by means of an even part of a
Weyl group , 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 .
The -orbit functions, determined by integral parameters, are invariant with
respect to even part of the affine Weyl group corresponding
to . The -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 of the
group (the discrete -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 . Such
functions can always be restricted without loss of information to a fundamental
region of the affine Weyl group. The members of each family satisfy
basic orthogonality relations when integrated over (continuous
orthogonality). It is demonstrated that the functions also satisfy discrete
orthogonality relations when summed up over a finite grid in
(discrete orthogonality), arising as the set of points in
representing the conjugacy classes of elements of a finite Abelian subgroup of
the maximal torus . The characters of the centre of the Lie
group allow one to split functions on into a sum
, where is the order of , and where the component
functions decompose into the series of -, or -, or -functions
from one congruence class only.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Orthogonal polynomials of compact simple Lie groups
Recursive algebraic construction of two infinite families of polynomials in
variables is proposed as a uniform method applicable to every semisimple
Lie group of rank . Its result recognizes Chebyshev polynomials of the first
and second kind as the special case of the simple group of type . The
obtained not Laurent-type polynomials are proved to be equivalent to the
partial cases of the Macdonald symmetric polynomials. Basic relation between
the polynomials and their properties follow from the corresponding properties
of the orbit functions, namely the orthogonality and discretization. Recurrence
relations are shown for the Lie groups of types , , , ,
, , and together with lowest polynomials.Comment: 34 pages, some minor changes were done, to appear in IJMM
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