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## Homework Statement

Show that vector representation

**5**and adjoint representation

**10**in SO(5) decompose respectively into representations of SO(4) as:

**5**→

**4**⊕

**1**

10→

10

**6**⊕

**4**

## Homework Equations

## 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?

**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)