arXiv subjects

J. Patera

Publications and source records attributed to J. Patera.

15 recordsLinked to original sources

arXiv2011

Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations

In the paper we present a new, uniform and comprehensive description of centralizers of the maximal regular subgroups in compact simple Lie groups of all types and ranks. The centralizer is either a direct product of finite cyclic groups, a continuous group of rank 1, or a product, not necessarily direct, of a continuous group of rank 1 with a finite cyclic group. Explicit formulas for the action of such centralizers on irreducible representations of the simple Lie algebras are given.

math-ph↗

arXiv2011

Branching rules for Weyl group orbits of simple Lie algebras B(n), C(n) and D(n)

The orbits of Weyl groups W(B(n)), W(C(n)) and W(D(n)) of the simple Lie algebras B(n), C(n) and D(n) are reduced to the union of the orbits of Weyl groups of the maximal reductive subalgebras of B(n), C(n) and D(n). Matrices transforming points of W(B(n)), W(C(n)) and W(D(n)) orbits into points of subalgebra orbits are listed for all cases n<=8 and for the infinite series of algebra-subalgebra pairs B(n) - B(n-1) x U(1), B(n) - D(n), B(n) - B(n-k) x D(k), B(n) - A(1), C(n) - C(n-k) x C(k), C(n) - A(n-1) x U(1), D(n) - A(n-1) x U(1), D(n) - D(n-1) x U(1), D(n) -B(n-1), D(n) - B(n-k-1) x B(k), D(n) -D(n-k) x D(k). Numerous special cases and examples are shown.

math-ph↗

arXiv2009

Orbit functions of SU(n) and Chebyshev polynomials

Orbit functions of a simple Lie group/Lie algebra L consist of exponential functions summed up over the Weyl group of L. They are labeled by the highest weights of irreducible finite dimensional representations of L. They are of three types: C-, S- and E-functions. Orbit functions of the Lie algebras An, or equivalently, of the Lie group SU(n+1), are considered. First, orbit functions in two different bases - one orthonormal, the other given by the simple roots of SU(n) - are written using the isomorphism of the permutation group of n elements and the Weyl group of SU(n). Secondly, it is demonstrated that there is a one-to-one correspondence between classical Chebyshev polynomials of the first and second kind, and C- and $S$-functions of the simple Lie group SU(2). It is then shown that the well-known orbit functions of SU(n) are straightforward generalizations of Chebyshev polynomials to n-1 variables. Properties of the orbit functions provide a wealth of properties of the polynomials. Finally, multivariate exponential functions are considered, and their connection with orbit functions of SU(n) is established.

math.CA↗

arXiv2009

Branching rules for the Weyl group orbits of the Lie algebra A(n)

The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra orbits are listed for all cases n<=8 and for the infinite series of algebra-subalgebra pairs A(n) - A(n-k-1) x A(k) x U(1), A(2n) - B(n), A(2n-1) - C(n), A(2n-1) - D(n). Numerous special cases and examples are shown.

math-ph↗

arXiv2009

The rings of n-dimensional polytopes

Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.

math.MG↗

arXiv2008

Computing with almost periodic functions

The paper develops a method for discrete computational Fourier analysis of functions defined on quasicrystals and other almost periodic sets. A key point is to build the analysis around the emerging theory of quasicrystals and diffraction in the setting on local hulls and dynamical systems. Numerically computed approximations arising in this way are built out of the Fourier module of the quasicrystal in question, and approximate their target functions uniformly on the entire infinite space. The methods are entirely group theoretical, being based on finite groups and their duals, and they are practical and computable. Examples of functions based on the standard Fibonacci quasicrystal serve to illustrate the method (which is applicable to all quasicrystals modeled on the cut and project formalism).

math-ph↗

arXiv2008

Three dimensional C-, S- and E-transforms

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-, $S$-, and $E$-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region $F$ of the 3-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid $F_M\subset F$ (discrete orthogonality). The positive integer $M$ sets up the density of the lattice containing $F_M$. The expansion of functions given either on $F$ or on $F_M$ is the paper's main focus.

math-ph↗

arXiv2007

(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family, allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms, involving the symmetric and antisymmetric multivariate sine and cosine functions.

math.CA↗

arXiv2007

(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are eigenfunctions of the Laplace operator on corresponding fundamental domains satisfying certain boundary conditions. To symmetric and antisymmetric multivariate exponential functions there correspond Fourier transforms. There are three types of such Fourier transforms: expansions into corresponding Fourier series, integral Fourier transforms, and multivariate finite Fourier transforms. Eigenfunctions of the integral Fourier transforms are found.

math.CA↗

arXiv2006

Graded contractions of the Pauli graded sl(3,C)

The Lie algebra $sl(3,\C)$ is considered in the basis of generalized Pauli matrices. Corresponding grading is the Pauli grading here. It is one of the four gradings of the algebra which cannot be further refined. The set $\es$ of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of $sl(3,\C)$, which is essentially the finite group $SL(2,\Z_3)$, and the induced symmetry of the system $\es$. A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classes are non-parametric and 13 solutions depend on one or two continuous parameters, providing a continuum of equivalence classes and subsequently continuum of non-isomorphic Lie algebras. Solutions of the contraction equations of Pauli graded $sl(3,\bC)$ are identified here as specific solvable Lie algebras of dimensions up to 8. Earlier algorithms for identification of Lie algebras, given by their structure constants, had to be made more efficient in order to distinguish non-isomorphic Lie algebras encountered here. Resulting Lie algebras are summarized in tabular form. There are 88 indecomposable solvable Lie algebras of dimension 8, 77 of them being nilpotent. There are 11 infinite sets of parametric Lie algebras which still deserve further study.

math-ph↗

arXiv2004

The fine gradings of sl(3,C) and their symmetries

We describe the normalizers for all non-conjugate maximal Abelian subgroups of diagonalizable automorphisms of sl(3,C) and show their relation to the symmetries of equations related to the graded contraction.

math-ph↗

arXiv2004

Fourier transform method for imaging atmospheric Cherenkov telescopes

We propose Fourier transform (FT) method for processing images of extensive air showers (EAS) detected by imaging atmospheric Cherenkov telescopes (IACT) used in the very high energy (VHE) gamma-ray astronomy. The method is based on the discrete Fourier transforms (DFT) on compact Lie groups, and the use of continuous extension of the inverse discrete transforms to approximate the discrete EAS images by continuous EAS brightness distribution functions. Here we describe the FT-method in case of SU(3) group. It allows practical realization of the DFT technique for functions sampled on hexagonal symmetry grids. The proposed method can also be implemented in case of IACT cameras with grids of rectangular symmetry by using the DFT on the SU(2)xSU(2) group. The proposed FT-method is applied to the Monte-Carlo simulated bank of TeV proton and gamma-ray EAS images for a stand-alone telescope. Comparing between the FT-method and the currently used standard method shows that the FT technique allows a better and systematic enhancement of the gamma-ray signal. The relative difference between these two methods becomes more profound especially for `photon poor' images. It suggests that the EAS detection thresholds of IACTs could be effectively reduced with the use of FT technique. This prediction is further supported by a significant noise suppression capability of the method using simple low-pass filters in the image frequency domain. The FT-method allows very deep `tail' (and `height') image cuts, differentiation of images, operations in the image frequency domain, etc., that can be used for development of new effective parameters for the EAS image processing.

astro-ph↗

arXiv2003

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 $Π_n$ 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 $Π_n$ 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 $Π_3$ 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 $Π_3$.

math-ph↗

arXiv2003

Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine transform (DCT), which is often considered to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of these two discrete transforms from the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are considered. (A) Unlike the continuous extension of the DFT, the continuous extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$ between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT converges to the derivative of $g(t)$. And (C), for CEDCT the principle of locality is valid. Finally, we use the continuous extension of 2-dimensional DCT to illustrate its potential for interpolation, as well as for the data compression of 2D images.

math-ph↗

arXiv2002

Affine extension of noncrystallographic Coxeter groups and quasicrystals

Unique affine extensions $H^{\aff}_2$, $H^{\aff}_3$ and $H^{\aff}_4$ are determined for the noncrystallographic Coxeter groups $H_2$, $H_3$ and $H_4$. They are used for the construction of new mathematical models for quasicrystal fragments with 10-fold symmetry. The case of $H^{\aff}_2$ corresponding to planar point sets is discussed in detail. In contrast to the cut-and-project scheme we obtain by construction finite point sets, which grow with a model specific growth parameter.

math.GR↗