Linear Representations of Groups 1989
DOI: 10.1007/978-3-0348-9274-2_4
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Representations of Compact Groups

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“…Noting that 1 and 1 ⊥ , the space of trace-0 Hermitian operators, are both invariant under this action, it follows that the two are orthogonal with respect to any unitarily invariant inner product on V * (F). Also, since the adjoint representation of U (H) on 1 ⊥ is irreducible ( [25], p. 20), it follows from Schur's Lemma that up to normalization, there is only one unitarily invariant inner product on the latter -in other words, any invariant inner product on 1 ⊥ has the form a, b = λ n Tr(ab) for some λ > 0, with λ = 1 corresponding to the normalized trace inner product. Hence, an invariant inner product on V = 1 ⊕ 1 ⊥ is entirely determined by the normalization of 1 and the choice of λ.…”
Section: Appendix B: An Alternative Route To Homogeneitymentioning
confidence: 99%
“…Noting that 1 and 1 ⊥ , the space of trace-0 Hermitian operators, are both invariant under this action, it follows that the two are orthogonal with respect to any unitarily invariant inner product on V * (F). Also, since the adjoint representation of U (H) on 1 ⊥ is irreducible ( [25], p. 20), it follows from Schur's Lemma that up to normalization, there is only one unitarily invariant inner product on the latter -in other words, any invariant inner product on 1 ⊥ has the form a, b = λ n Tr(ab) for some λ > 0, with λ = 1 corresponding to the normalized trace inner product. Hence, an invariant inner product on V = 1 ⊕ 1 ⊥ is entirely determined by the normalization of 1 and the choice of λ.…”
Section: Appendix B: An Alternative Route To Homogeneitymentioning
confidence: 99%
“…In general, the presented material contributes to the creation and development of new approaches in various fields of Newtonian mechanics, in the branches of classical continuum mechanics. These new approaches are based on fundamental achievements of the science of mechanics and mathematics [52][53][54][55][56][57][58][59][60][61][62][63][64]. They were carried out by the author [65][66][67][68][69][70][71][72][73] and were realized by his colleagues in different branches: In elasto-plasticity at finite strains [74,75], in constructing models of Cosserat type [75][76][77][78][79], in poromechanics [79,80], in the theory of shape-memory materials [81], in the generalization of the theory of elastic-plastic processes to finite deformations [82], in numerical methods for elastic-plastic problems at finite strains [83], in analytical research in hypo-elasticity [84], and in constructing new models of visco-elasticity at finite strains by using new methods, including the application of different objective derivatives [85][86][87].…”
mentioning
confidence: 99%