A finite subgroup of the exceptional Lie group<i>G</i><sub>2</sub>

King, Ronald C.; Toumazet, Frédéric; Wybourne, B. G.

doi:10.1088/0305-4470/32/48/312

Cited by 15 publications

(29 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The irreps of the continuous groups of interest, such as W(abc) for SO (7) or U(ab) for G 2 , can be depicted as Table 1. Terms for the d n configurations (0 n 5) and the corresponding irreps of SO(5).…”

Section: Irreps Of Continuous Groups: Geometrical Modelsmentioning

confidence: 99%

“…Thus a fivedimensional D spherical harmonic appears with a ninedimensional G spherical harmonic in the (200) irrep of SO (7) or the (20) irrep of G 2 for a total dimension of 14 (table 2). Similarly the three-dimensional P spherical harmonic appears with an eleven-dimensional H spherical harmonic in the (110) irrep of SO(7) or the (11) irrep of G 2 , again for a total dimension of 14.…”

Section: 4mentioning

confidence: 99%

“…The relevant group is the infinite group SO (7), which is the full rotation group in seven dimensions corresponding to the seven orbital states of an f electron. In this work, Racah already recognized that the exceptional Lie group G 2 was a useful subgroup of SO (7) for studying the f n electron configurations.…”

Section: Introductionmentioning

confidence: 99%

“…The key finite subgroup of SO (7) for the study of the atomic f shell would seem to be the heptakisoctahedral [4] or didodecahedral [5] group, 7 O, of order 168. This group can be constructed from a finite field of order 7 in a manner exactly analogous to the construction of the icosahedral group from a finite field of order 5 [6].…”

Section: Introductionmentioning

confidence: 99%

“…This group can be constructed from a finite field of order 7 in a manner exactly analogous to the construction of the icosahedral group from a finite field of order 5 [6]. Furthermore 7 O is a finite subgroup of G 2 [7] leading to the subgroup chain SO(7) ' G 2 ' 7 O for the study of the atomic f shell analogous to the subgroup chain SO(5) ' SO(3) ' I for the study of the atomic d shell.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

King

2006

Molecular Physics

View full text Add to dashboard Cite

This paper discusses the use of the non-Euclidean heptakisoctahedral group, 7 O, of order 168, to study the atomic f shell of interest in lanthanide and actinide chemistry. In this group only the representations of the spherical harmonics of dimensions 0, 1, or 6 (mod 7) can be reduced to sums of irreducible representations (irreps). These include the S, F, I, K, and N spherical harmonics with ' values of 0, 3, 6, 7, and 10, respectively. The other spherical harmonics must be combined into pairs with total dimension a multiple of 7 in order to have representations reducible into sums of irreps of 7 O. Such pairs of interest in the atomic f shell include the PH and DG pairs of total dimension 14, the GM and HL pairs of total dimension 28, and the LQ and MO pairs of total dimension 42. If the subgroup chain SO(7) ' G 2 ' 7 O is used to study the atomic f shell, terms corresponding to these paired spherical harmonics, such as PH and DG pairs found in the highest spin f n configurations, appear together under a given irrep of SO(7) or G 2 .

show abstract

Section: Irreps Of Continuous Groups: Geometrical Modelsmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

King

2006

Molecular Physics

View full text Add to dashboard Cite

show abstract

Supersymmetric M2-branes with Englert fluxes, and the simple group PSL(2, 7)

Fré

2016

Fortschr. Phys.

View full text Add to dashboard Cite

A new class is introduced of M2-branes solutions of d=11 supergravity that include internal fluxes obeying Englert equation in 7-dimensions. A simple criterion for the existence of Killing spinors in such backgrounds is established. Englert equation is viewed as the generalization to d=7 of Beltrami equation defined in d=3 and it is treated accordingly. All 2-brane solutions of minimal d=7 supergracity can be uplifted to d=11 and have N ≥ 4 supersymmetry. It is shown that the simple group PSL(2, 7) is crystallographic in d=7 having an integral action on the A7 root lattice. By means of this point-group and of the T 7 torus obtained quotiening R 7 with the A7 root lattice we were able to construct new M2 branes with Englert fluxes and N ≤ 4. In particular we exhibit here an N = 1 solution depending on 4-parameters and admitting a large non abelian discrete symmetry, namely G 21 ≡ Z 3 Z 7 ⊂ PSL(2, 7). The dual d = 3 field theories have the same symmetries and have complicated non linear interactions.

show abstract

The Integral Form of D = 3 Chern‐Simons Theories Probing Singularities

Fré

Grassi

2017

Fortschritte der Physik

View full text Add to dashboard Cite

We consider D=3 supersymmetric Chern Simons gauge theories both from the point of view of their formal structure and of their applications to the AdS4/CFT3 correspondence. From the structural view‐point, we use the new formalism of integral forms in superspace that utilizes the rheonomic Lagrangians and the Picture Changing Operators, as an algorithmic tool providing the connection between different approaches to supersymmetric theories. We provide here the generalization to an arbitrary Kähler manifold with arbitrary gauge group and arbitrary superpotential of the rheonomic lagrangian of D=3 matter coupled gauge theories constructed years ago. From the point of view of the AdS4/CFT3 correspondence and more generally of M2‐branes we emphasize the role of the Kähler quotient data in determining the field content and the interactions of the Cherns Simons gauge theory when the transverse space to the brane is a non‐compact Kähler quotient K4 of some flat variety with respect to a suitable group. The crepant resolutions of Cn/Γ singularities fall in this category. In the present paper we anticipate the general scheme how the geometrical data are to be utilized in the construction of the D=3 Chern‐Simons Theory supposedly dual to the corresponding M2‐brane solution.

show abstract

A finite subgroup of the exceptional Lie groupG₂

Cited by 15 publications

References 10 publications

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

Supersymmetric M2-branes with Englert fluxes, and the simple group PSL(2, 7)

The Integral Form of D = 3 Chern‐Simons Theories Probing Singularities

Contact Info

Product

Resources

About

A finite subgroup of the exceptional Lie groupG2

Cited by 15 publications

References 10 publications

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

Supersymmetric M2-branes with Englert fluxes, and the simple group PSL(2, 7)

The Integral Form of D = 3 Chern‐Simons Theories Probing Singularities

Contact Info

Product

Resources

About

A finite subgroup of the exceptional Lie groupG₂