scholarly journals Infinite irreducible representations of the Lorentz group

1947 ◽  
Vol 189 (1018) ◽  
pp. 372-401 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Irreducible Representation ◽  
Lorentz Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Spin Properties

It is shown that corresponding to every pair of complex numbers κ , κ* for which 2( κ - κ* ) is real and integral, there exists, in general, one irreducible representation D κ, κ* , of the Lorentz group. However, if 4 κ , 4 κ* are both real and integral there are two representations D + κ, κ* and D - k, k* associated to the pair ( k, κ* ). All these representations are infinite except D - κ, κ* which is finite if 2 κ , 2 κ* are both integral. For suitable values of ( κ, κ* ), D κ, κ* or D + κ, κ* is unitary. U and B matrices similar to those given by Dirac (1936) and Fierz (1939) are introduced for these infinite representations. The extension of Dirac’s expansor formalism to cover half-integral spins is given. These new quantities, which are called expinors, bear the same relation to spinors as Dirac’s expansors to tensors. It is shown that they can be used to describe the spin properties of a particle in accordance with the principles of quantum mechanics.

On Springer's correspondence for simple groups of type En (n = 6, 7, 8)

1982 ◽  
Vol 92 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Dean Alvis ◽  
George Lusztig
Keyword(s):  
Irreducible Representation ◽  
Tensor Product ◽  
Algebraic Group ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Complex Numbers ◽  
Unipotent Element ◽  
Reductive Algebraic Group ◽  
Dimension 2 ◽  
Injective Map

Let G be a connected reductive algebraic group over complex numbers. To each unipotent element u ε G (up to conjugacy) and to the unit representation of the group of components of the centralizer of u, Springer (11), (12) associates an irreducible representation of the Weyl group W of G. The tensor product of that representation with the sign representation will be denoted ρu. (This agrees with the notation of (5).) This representation may be realized as a subspace of the cohomology in dimension 2β(u) of the variety of Borel subgroups containing u, where β(u) = dim . For example, when u = 1, ρu is the sign representation of W. The map u → ρu defines an injective map from the set of unipotent conjugacy classes in G to the set of irreducible representations of W (up to isomorphism). Our purpose is to describe this map in the case where G is simple of type Eu (n = 6, 7, 8). (When G is classical or of type F4, this map is described by Shoji (9), (10); the case where G is of type G2 is contained in (11).

Restriction of irreducible representations of groups to a subgroup

1971 ◽  
Vol 322 (1549) ◽  
pp. 181-189 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Irreducible Representation ◽  
Symmetry Breaking ◽  
Irreducible Representations ◽  
Representations Of Groups ◽  
Group Play ◽  
Special Case

The number and character of the irreducible representations of a subgroup, contained in an irreducible representation of the whole group (if this representation is restricted to the sub­ group) play an important role in quantum mechanics. They give the number and type of the states, generated by a symmetry breaking perturbation, from a state which has the symmetry of the whole group. Three equations are derived here for the number and character of the representations of the subgroup, resulting from the restriction of the irreducible represen­tations of the whole group. These equations contain an earlier rule as a special case.

On The Number of Classes of a Finite Group Invariant for Certain Substitutions

1974 ◽  
Vol 26 (5) ◽  
pp. 1090-1097 ◽  
Author(s):  
A. J. van Zanten ◽  
E. de Vries
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Symmetric Group ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Group Invariant ◽  
A Finite Group ◽  
Number Of Classes ◽  
Special Case

In this paper we consider representations of groups over the field of the complex numbers.The nth-Kronecker power σ⊗n of an irreducible representation σ of a group can be decomposed into the constituents of definite symmetry with respect to the symmetric group Sn. In the special case of the general linear group GL(N) in N dimensions the decomposition of the defining representation at once provides irreducible representations of GL(N) [9; 10; 11].

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

scholarly journals REFINING THE CLASSIFICATION OF THE IRREPS OF THE 1D N-EXTENDED SUPERSYMMETRY

2008 ◽  
Vol 23 (01) ◽  
pp. 37-51 ◽  
Author(s):  
ZHANNA KUZNETSOVA ◽  
FRANCESCO TOPPAN
Keyword(s):  
Quantum Mechanics ◽  
Extended Supersymmetry ◽  
Supersymmetric Quantum Mechanics ◽  
Irreducible Representations

The linear finite irreducible representations of the algebra of the 1D N-Extended Supersymmetric Quantum Mechanics are discussed in terms of their "connectivity" (a symbol encoding information on the graphs associated to the irreps). The classification of the irreducible representations with the same fields content and different connectivity is presented up to N ≤ 8.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

scholarly journals Dimensions of the Irreducible Representations of the Symmetric and Alternating Group

10.37236/4909 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Korneel Debaene
Keyword(s):  
Irreducible Representation ◽  
Irreducible Representations ◽  
Alternating Group ◽  
Main Ingredient ◽  
Short Intervals ◽  
Prime Factors

We establish the existence of an irreducible representation of $A_n$ whose dimension does not occur as the dimension of an irreducible representation of $S_n$, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in the proof is a result on large prime factors in short intervals. 

Sign in / Sign up

Export Citation Format

Share Document