Irreducible representations of the exceptional lie groups
Irreducible representations of the exceptional lie groups
Some properties of the exceptional Lie groups G2, F,,, Es, E7 and Es and of their irreducible representations are investigated. In each case the roots of the corresponding Lie algebra are chosen in such a way that a maximally embedded classical Lie algebra of the same rank is made manifest. In the same way the irreducible representations of these exceptional Lie groups are labelled in a manner as close as possible to that used for the classical Lie groups. The corresponding .characters obtained from Weyl's formula are simplified to the extent that they are expressed explicitly by means of formulae involving only permutations and sign changes rather than the complete set of Weyl operations. In the case of G2 this yields a very simple statement of the branching rule for the restrictions of SO(7) to G2 and of G2 to SU(3). Explicit formulae are given for the branching multiplicities in both cases, and the second of these is used to evaluate the weight multiplicities of G2 ..The same technique is used to derive the branching rule appropriate to the restriction of F4 to SO(9). The rule is reduced to a formula involving the known Kroneckar products of irreducible representations of SO(9). Comparison is made with the powerful method of elementary multiplets involving stretched products. The ki.own weight multiplicities of SO(9) are then used to evaluate the weight multiplicities of F4 .The results are then extended to the groups Es , E7 and Es for which the corresponding Weyl groups are discussed in detail. The maximally embedded classical Lie subgroups employed in this analysis areSU(2) x SU(6) , SU(8) and SO(16) respectively. The Weyl group of the exceptional Lie groups is used to generate a set of dominant weights of the classical subgroup related to each dominant weight of the exceptional group. The corresponding branching rules are then used to evaluate and tabulate the dominant weight multiplicities of Es ,E7 and Es.Some discussion is also given of the Krocker products of irreducible representation of the exceptional Lie groups. The method used employs the Weyl group once again, this time to generate the modification rules needed to deal with inadmissible irreducible representation.
University of Southampton
al-Qubanshi, Aladdin Hassan Ali
1978
al-Qubanshi, Aladdin Hassan Ali
al-Qubanshi, Aladdin Hassan Ali
(1978)
Irreducible representations of the exceptional lie groups.
University of Southampton, Doctoral Thesis.
Record type:
Thesis
(Doctoral)
Abstract
Some properties of the exceptional Lie groups G2, F,,, Es, E7 and Es and of their irreducible representations are investigated. In each case the roots of the corresponding Lie algebra are chosen in such a way that a maximally embedded classical Lie algebra of the same rank is made manifest. In the same way the irreducible representations of these exceptional Lie groups are labelled in a manner as close as possible to that used for the classical Lie groups. The corresponding .characters obtained from Weyl's formula are simplified to the extent that they are expressed explicitly by means of formulae involving only permutations and sign changes rather than the complete set of Weyl operations. In the case of G2 this yields a very simple statement of the branching rule for the restrictions of SO(7) to G2 and of G2 to SU(3). Explicit formulae are given for the branching multiplicities in both cases, and the second of these is used to evaluate the weight multiplicities of G2 ..The same technique is used to derive the branching rule appropriate to the restriction of F4 to SO(9). The rule is reduced to a formula involving the known Kroneckar products of irreducible representations of SO(9). Comparison is made with the powerful method of elementary multiplets involving stretched products. The ki.own weight multiplicities of SO(9) are then used to evaluate the weight multiplicities of F4 .The results are then extended to the groups Es , E7 and Es for which the corresponding Weyl groups are discussed in detail. The maximally embedded classical Lie subgroups employed in this analysis areSU(2) x SU(6) , SU(8) and SO(16) respectively. The Weyl group of the exceptional Lie groups is used to generate a set of dominant weights of the classical subgroup related to each dominant weight of the exceptional group. The corresponding branching rules are then used to evaluate and tabulate the dominant weight multiplicities of Es ,E7 and Es.Some discussion is also given of the Krocker products of irreducible representation of the exceptional Lie groups. The method used employs the Weyl group once again, this time to generate the modification rules needed to deal with inadmissible irreducible representation.
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Published date: 1978
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Local EPrints ID: 458639
URI: http://eprints.soton.ac.uk/id/eprint/458639
PURE UUID: 1af0bd40-b7e9-473a-ab4f-280265abb77a
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Date deposited: 04 Jul 2022 16:52
Last modified: 04 Jul 2022 16:52
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Author:
Aladdin Hassan Ali al-Qubanshi
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