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Representation decomposition
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[QUOTE="Augbrah, post: 5475166, member: 593705"] [h2]Homework Statement [/h2] Show that vector representation [B]5 [/B]and adjoint representation [B]10[/B] in SO(5) decompose respectively into representations of SO(4) as: [B]5[/B] →[B]4[/B]⊕[B]1 10[/B]→[B]6[/B]⊕[B]4[/B] [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] [/B] I understand that [B]5 [/B]is rep of SO(5) corresponding to Dynkin labels (1, 0). [B]1[/B] is of course trivial representation. But what type of rep is [B]4[/B]? Once I started calculating weights I found out that Dynkin labels corresponding to (2, 0) as well as (1, 1) for SO(4) both have 4 weights (and different ones). And none of them seem to reproduce [B]5![/B] Maybe there is an easier way, how would you show these decomposition? [SPOILER="Results I've got"]Weights for [B]5 [/B][Dynkin (1,0) for SO(5)]: (1, 0), (-1, 2), (0,0), (1, -2) (0, -1) Weights for [B]4? [/B][Dynkin (1,1) for SO(4)]: (1,1), (-1, 1), (1, -1), (-1, -1) Weights for [B]4? [/B][Dynkin (3, 0) for SO(4)]: (3, 0), (1, 0), (-1, 0), (-3, 0) [/SPOILER] [/QUOTE]
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Representation decomposition
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