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.

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 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.

scholarly journals Spontaneous symmetry breaking in quantum mechanics

2007 ◽  
Vol 75 (7) ◽  
pp. 635-638 ◽  
Author(s):  
Jasper van Wezel ◽  
Jeroen van den Brink
Keyword(s):  
Quantum Mechanics ◽  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking

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. 

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Irreducible representations of E theory

2019 ◽  
Vol 34 (24) ◽  
pp. 1950133 ◽  
Author(s):  
Peter West
Keyword(s):  
Irreducible Representation ◽  
Direct Product ◽  
Simple Form ◽  
Light Cone ◽  
Degrees Of Freedom ◽  
Irreducible Representations ◽  
Vector Representation ◽  
Maximal Supergravity ◽  
Duality Relations ◽  
Cartan Involution

We construct the [Formula: see text] theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of [Formula: see text] with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in [Formula: see text] theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of [Formula: see text]. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

Gauge hierarchies for chiral SU(N) × SU(N) groups

1976 ◽  
Vol 54 (16) ◽  
pp. 1660-1663 ◽  
Author(s):  
Shalom Eliezer
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Gauge Theories ◽  
Vacuum Expectation ◽  
Coupling Constants ◽  
Spontaneous Breaking ◽  
Text Representation ◽  
Algebraic Equations ◽  
Expectation Values ◽  
Special Case

We have presented a special case where a hierarchy of spontaneous breaking of the symmetries can be achieved in conventional gauge theories (i.e. the Higgs scalars are elementary bosons and the coupling constants of the quartic interactions are small). We break spontaneously the chiral group SU(N) × SU(N) with Higgs scalars transforming like the (N, [Formula: see text]) representation of SU(N) × SU(N). By minimizing the potential we obtain a set of algebraic equations of the type[Formula: see text]where ηj are the vacuum expectation values of the Higgs scalars and μi2 and Aij are parameters. In order to get a hierarchy of spontaneous symmetry breaking we obtain the condition det Aij = 0.

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