Basis states for equivalent electrons. IV. States for the Lie group SO(2l + 1) × SU(2) using U(2l + 1) × SU(2) states

1981 ◽  
Vol 59 (3) ◽  
pp. 315-324
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Irreducible Representations ◽  
Intermediate Group ◽  
Slater Basis

In this paper we examine the irreducible representations of SO(2l + 1) × SU(2) that are present when we have equivalent electrons. Basis states for these irreducible representations are defined using the basis states for U(2l + 1) × SU(2). Using earlier results, where the U(2l + 1) × SU(2) states were defined using the U(4l + 2) states, we are able to express the SO(2l + 1) × SU(2) states in terms of the Slater basis states associated with the irreducible representation (1N) of U(4l + 2). The SO(2l + 1) × SU(2) states obtained in this paper using the intermediate group U(2l + 1) × SU(2) are compared with those obtained earlier using the intermediate group Sp(4l + 2).

Basis states for equivalent electrons. II. States for the Lie group Sp(4l + 2)

1980 ◽  
Vol 58 (12) ◽  
pp. 1724-1728
Author(s):  
William R. Ross
Keyword(s):  
Angular Momentum ◽  
Irreducible Representation ◽  
Orbital Angular Momentum ◽  
Lie Group ◽  
Total Spin ◽  
Irreducible Representations ◽  
Total Orbital Angular Momentum ◽  
Intermediate Group ◽  
Slater Basis

The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of the Lie group U(4l + 2). States which are eigenfunctions of the total spin and total orbital angular momentum form the basis for irreducible representations of SO(3) × SU(2). In this paper the intermediate group Sp(4l + 2) is studied. The basis states for irreducible representations of Sp(4l + 2) are expressed in terms of the Slater basis states.

Basis states for equivalent electrons. III. States for the Lie group SO(2l + 1) × SU(2) using Sp(4l + 2) states

1981 ◽  
Vol 59 (2) ◽  
pp. 207-212
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Unitary Group ◽  
Symplectic Group ◽  
Irreducible Representations ◽  
Slater Basis

The Lie group SO(2l + 1) × EU(2) is a subgroup of the symplectic group Sp(4l + 2), which in turn is a subgroup of the unitary group U(4l + 2). The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of U(4l + 2). The basis states for the irreducible representations of SO(2l + 1) × SU(2) are expressed in terms of the states for irreducible representations of Sp(4l + 2). The basis states for SO(2l + 1) × SU(2) are also expressed in terms of the Slater basis states.

Basis states for equivalent electrons. I. States for the Lie group U(2l + 1) × SU(2)

1980 ◽  
Vol 58 (12) ◽  
pp. 1718-1723
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Spin Operator ◽  
Total Spin ◽  
Slater Basis

The antisymmetric Slater basis states for N equivalent electrons form the basis for an irreducible representation of U(4l + 2). When we consider the subgroup U(2l + 1) × SU(2) we obtain states which are eigenstates of the total spin operator. The basis states for the irreducible representation of U(2l + 1) × SU(2) are expressed in terms of the Slater basis states. General expressions are obtained which can easily be applied regardless of the number of electrons, the value of l, or the irreducible representation that is considered.

HARDY’S THEOREM FOR GABOR TRANSFORM

2018 ◽  
Vol 106 (2) ◽  
pp. 143-159
Author(s):  
ASHISH BANSAL ◽  
AJAY KUMAR ◽  
JYOTI SHARMA
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Discrete Group ◽  
Irreducible Representations ◽  
Gabor Transform ◽  
Nilpotent Lie Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Hardy’S Theorem ◽  
Compact Abelian Groups

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once

1972 ◽  
Vol 24 (3) ◽  
pp. 432-438 ◽  
Author(s):  
Fredric E. Goldrich ◽  
Eugene P. Wigner
Keyword(s):  
Irreducible Representation ◽  
Lie Groups ◽  
Finite Groups ◽  
Lie Group ◽  
Unitary Group ◽  
Unit Vector ◽  
Doctoral Dissertation ◽  
Irreducible Representations ◽  
Compact Lie Group ◽  
Compact Lie Groups

One of the results of the theory of the irreducible representations of the unitary group in n dimensions Un is that these representations, if restricted to the subgroup Un-1 leaving a vector (let us say the unit vector e1 along the first coordinate axis) invariant, do not contain any irreducible representation of this Un-1 more than once (see [1, Chapter X and Equation (10.21)]; the irreducible representations of the unitary group were first determined by I. Schur in his doctoral dissertation (Berlin, 1901)). Some time ago, a criterion for this situation was derived for finite groups [3] and the purpose of the present article is to prove the aforementioned result for compact Lie groups, and to apply it to the theory of the representations of Un.

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 A remark on the tensor product of irreducible representations of a compact Lie group

1981 ◽  
Vol 7 (1) ◽  
pp. 201-211
Author(s):  
C. FRONSDAL ◽  
T. HIRAI
Keyword(s):  
Tensor Product ◽  
Lie Group ◽  
Irreducible Representations ◽  
Compact Lie Group

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).

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