scholarly journals REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250019
Author(s):  
PHILIP FEINSILVER
Keyword(s):  
Special Functions ◽  
Regular Representation ◽  
Boolean Lattice ◽  
Irreducible Representations ◽  
Leibniz Rule ◽  
Krawtchouk Polynomials ◽  
Lattice Representation ◽  
Boolean Poset ◽  
Group Law

Starting with the zero-square "zeon algebra," the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard–Sylvester matrices are shown as well.

scholarly journals On new applications of fractional calculus

2017 ◽  
Vol 37 (3) ◽  
pp. 113-118 ◽  
Author(s):  
Shilpi Jain ◽  
Praveen Agarwal
Keyword(s):  
Fractional Calculus ◽  
Special Functions ◽  
Leibniz Rule ◽  
The One ◽  
New Applications ◽  
Paper Author

In the present paper author derive a number of integrals concerning various special functions which are applications of the one of Osler result. Osler provided extensions to the familiar Leibniz rule for the nth derivative of product of two functions.

scholarly journals On the Structure of Finite Groupoids and Their Representations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 414 ◽  
Author(s):  
Alberto Ibort ◽  
Miguel Rodríguez
Keyword(s):  
Irreducible Representation ◽  
Regular Representation ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Linear Representations ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Quantum Mechanical Systems ◽  
Maschke’S Theorem ◽  
Totally Disconnected

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

FINE DISINTEGRATION OF THE LEFT REGULAR REPRESENTATION

2005 ◽  
Vol 04 (06) ◽  
pp. 683-706 ◽  
Author(s):  
JEAN LUDWIG ◽  
CARINE MOLITOR-BRAUN
Keyword(s):  
Regular Representation ◽  
Unitary Irreducible Representation ◽  
Irreducible Representations ◽  
Left Translation ◽  
Direct Integral ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Left Regular Representation ◽  
Left Regular

Let Hn be the (2n + 1)-dimensional Heisenberg group. We decompose L2(Hn) as the closure of a direct sum of infinitely many left translation invariant eigenspaces (for certain systems of partial differential equations). The restriction of the left regular representation to each one of these eigenspaces disintegrates into a direct integral of unitary irreducible representations, such that each infinite dimensional unitary irreducible representation appears with multiplicity 0 or 1 in this disintegration.

scholarly journals Fourier algebras of parabolic subgroups

2017 ◽  
Vol 120 (2) ◽  
pp. 272 ◽  
Author(s):  
Søren Knudby
Keyword(s):  
Regular Representation ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Irreducible Representations ◽  
Fourier Algebra ◽  
Real Rank ◽  
Parabolic Subgroups ◽  
Direct Sum ◽  
Rank One ◽  
Minimal Parabolic Subgroup

We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.

scholarly journals Irreducible Representations of Deformed Oscillator Algebra and q-Special Functions

1997 ◽  
Vol 12 (01) ◽  
pp. 153-158 ◽  
Author(s):  
E. V. Damaskinsky ◽  
P. P. Kulish
Keyword(s):  
Moment Problem ◽  
Special Functions ◽  
Hermite Polynomials ◽  
Irreducible Representations ◽  
Oscillator Algebra ◽  
Exponential Functions ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Classical Moment Problem

Different generators of a deformed oscillator algebra give rise to one-parameter families of q-exponential functions and q-Hermite polynomials. Connections of the Stieltjes and Hamburger classical moment problem with the corresponding resolution of unity are also pointed out.

scholarly journals ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

2016 ◽  
Vol 56 (6) ◽  
pp. 440-447
Author(s):  
Lenka Háková ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Special Functions ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
One Dimensional ◽  
Complex Functions ◽  
Group Orbit ◽  
Rank 2 ◽  
Definition Of

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

scholarly journals Certain Unitary Representations of the Infinite Symmetric Group, I

1987 ◽  
Vol 105 ◽  
pp. 121-128 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Regular Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Type I ◽  
Discrete Topology ◽  
Type Ii ◽  
Infinite Symmetric Group ◽  
Von Neumann ◽  
Group I

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

scholarly journals Fractional derivatives of certain special functions

2001 ◽  
Vol 32 (2) ◽  
pp. 103-109
Author(s):  
V. B. L. Chaurasia ◽  
Anju Godika
Keyword(s):  
General Class ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Leibniz Rule ◽  
Special Cases ◽  
Derivatives Of

The theorems relating to the fractional derivatives for the multivariable H-function [1,15,17] and a general class of multivariable polynomials [13] have been established in this paper. Use of well known generalized Leibniz rule has been made to derive some theorems. Certain special cases of the main theorems have also been discussed.

Sign in / Sign up

Export Citation Format

Share Document