Automorphisms of Order 2 of an Abelian Group - download pdf or read online

February 14, 2018 | Symmetry And Group | By admin | 0 Comments

By Miller G. A.

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Kn−1 The corresponding matrix element of the transpose of this product is [(A1 A2 · · · An )t ]ij = (A1 A2 · · · An )ji . ,kn−1 We conclude that (A1 A2 · · · An )t = Atn Atn−1 · · · At1 9 and, similarly, that (A1 A2 · · · An )† = A†n A†n−1 · · · A†1 Group Theory Problem Set 5 November 6, 2001 Note: Problems marked with an asterisk are for Rapid Feedback. 1. 2, we established the relation Bi Bi† = I. Using the definitions in that proof, show that this result implies that Bi† Bi = I as well. Hint: Show that Bi Bi† = I implies that A˜i DA˜†i = D.

How many irreducible representations are there and what are their dimensions? ∗ Construct the character table of this group by following the procedure outlined below: (a) Enter the characters for the identical and “parity” representations. As in the case of S3 , the characters for the parity representation are either +1 or −1, depending on whether or not the the operation preserves the “handedness” of the coordinate system. (b) Enter the characters for the “coordinate” representation obtained from the action on (x, y) for each group operation.

Similarly, the representation of σd,2 can also be determined directly by inspection: 1 0 , 0 −1 which yields the character χ(3σd ) = 0 . We can now add the entries for this two-dimensional irreducible representations to the character table of C6v : C6v Γ1 Γ1 Γ1 Γ1 Γ2 Γ2 E 1 1 1 1 2 2 2C6 1 1 1 2C3 1 1 −1 C2 1 1 −2 3σv 1 −1 3σd 1 −1 0 0 (c) The one-dimensional irreducible representations must obey the multiplication table, since they themselves are representations of the group. In particular, given the products C3 C32 = E, C33 = E , if we denote by α the character of the class 2C3 = {C3 , C32 }, then these products require that α2 = 1, α3 = 1 , 9 respectively.

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Automorphisms of Order 2 of an Abelian Group by Miller G. A.

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