Regular representation

Algebraic Aspects

Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019

Marlos A. G. Viana, Vasudevan Lakshminarayanan

The regular representation results from Dn acting on itself as a group, as introduced in Section 2.2, whereas in the permutation representation the action is on the ordinary set {1,2,…,n}. The regular representation of Dn is a subgroup of S2n, whereas its permutation representation is a subgroup of Sn. Both Sn and S2n are isomorphic to finite subgroups of the corresponding general linear group GL(ℝ).

On the impossibility of localised states for molecular rotors with cyclic potentials

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Journal Information

Published in Molecular Physics, 2018

Thomas Grohmann, Jörn Manz

Treating localised and symmetry-adapted wave functions within a matrix-vector approach also allows us to use an alternative method to show that localised states are superpositions of symmetry-adapted wave functions. Therefore, we make use of the regular representation , a key concept in the representation theory of groups [67,68]. The characters of this representation are Within this representation, every -dimensional irreducible representation of the group occurs just times. Hence, for a cyclic group, the regular representation contains all irreducible representations of the group exactly once. Conversely, if the representation that is spanned by the set of all localised states, , contains or is identical to the regular representation , each contorsional symmetry that is possible in does indeed occur. Consequently, if the functions form a basis for a representation that contains , the localised states are superpositions of symmetry-adapted states .

Regular representation

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Algebraic Aspects

On the impossibility of localised states for molecular rotors with cyclic potentials