# Representation decomposition

## Homework Statement

Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as:

541
10
64

## The Attempt at a Solution

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I understand that 5 is rep of SO(5) corresponding to Dynkin labels (1, 0). 1 is of course trivial representation. But what type of rep is 4? 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 5! Maybe there is an easier way, how would you show these decomposition?

Weights for 5 [Dynkin (1,0) for SO(5)]:
(1, 0), (-1, 2), (0,0), (1, -2) (0, -1)

Weights for 4? [Dynkin (1,1) for SO(4)]:
(1,1), (-1, 1), (1, -1), (-1, -1)

Weights for 4? [Dynkin (3, 0) for SO(4)]:
(3, 0), (1, 0), (-1, 0), (-3, 0)