The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartans method of moving frames.
The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartans method of moving frames and the special technique developed for triangular and related algebras in [J. Phys. A: Math. Theor. 40 (2007), 7557-7572]. The conjecture of Tremblay and Winternitz [J. Phys. A: Math. Gen. 34 (2001), 9085-9099] on the number and form of elements in the bases is completed and proved.
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.
Lie groups with two different root lengths allow two mixed sign homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.
Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for construction of the polynomials are presented. Orthogonality relations of the four families of polynomials are found together with the appropriate weight fuctions. Tables of the lowest degree polynomials are shown. Numerous trigonometric-like identities are found. Two of the four families of functions are identified as the functions encountered in the Weyl character formula for the finite dimensional irreducible representations of the compact Lie group Sp(4). The other two families of functions seem to play no role in Lie theory so far in spite of their analogous `good properties.
Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank not greater then 3 are explicitly studied. We derive the polynomials of simple Lie groups B_3 and C_3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.
The paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of $n$ variables based on the root lattices of compact simple Lie groups $G$ of any type and of rank $n$. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from $A$-type lattices, yield Gaussian cubature formulae for each simple Lie group $G$ based on interpolation points that arise from regular elements of finite order in $G$. The polynomials arise from the irreducible characters of $G$ and the interpolation points as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. 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 $A_1$, $A_2$, $A_3$, $C_2$, $C_3$, $G_2$, and $B_3$ together with lowest polynomials.
We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.
We define and study multivariate exponential functions, symmetric with respect to the alternating group A_n, which is a subgroup of the permutation (symmetric) group S_n. These functions are connected with multivariate exponential functions, determined as the determinants of matrices whose entries are exponential functions of one variable. Our functions are eigenfunctions of the Laplace operator. By means of alternating multivariate exponential functions three types of Fourier transforms are constructed: expansions into corresponding Fourier series, integral Fourier transforms, and multivariate finite Fourier transforms. Alternating multivariate exponential functions are used as a kernel in all these Fourier transforms. Eigenfunctions of the integral Fourier transforms are obtained.
