Representation decomposition

[Augbrah]

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→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?

Spoiler: Results I've got

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)

 

[mighty2000]

To show this decomposition, we can use the branching rules for SO(5) and SO(4). The branching rule for SO(5) → SO(4) is given by:

(1, 0) → (2, 0) ⊕ (1, 1)

This means that the representation (1, 0) of SO(5) can be decomposed into a representation (2, 0) and a representation (1, 1) of SO(4).

Using this rule, we can see that the representation (1, 0) of SO(5) decomposes into a representation (2, 0) of SO(4) and a trivial representation (1, 0) of SO(4). This is the same as the decomposition shown in the forum post, 5 → 4 ⊕ 1.

Similarly, the branching rule for SO(5) → SO(4) can be used to show that the representation (2, 0) of SO(5) decomposes into a representation (1, 1) of SO(4) and a representation (1, 0) of SO(4). This gives us the decomposition 10 → 6 ⊕ 4, as shown in the forum post.

In summary, to show these decompositions, we need to use the branching rules for SO(5) → SO(4) and the fact that the representation (1, 0) of SO(5) corresponds to the vector representation, while the representation (2, 0) corresponds to the adjoint representation.