Irreducibility of (anti)self-dual reps

  • A
  • Thread starter gentsagree
  • Start date
  • #1
96
1
Imagine we are talking about the group SO(4). The second rank antisymmetric representation is reducible into self-dual and antiself-dual representations. I think a good way to visualise this is by noticing that the projection of [itex] \Lambda^{2}V [/itex] into self and antiself dual subspaces commutes with the action of SO(4).

However, how can I show that those subspaces are themselves irreducible?

Thanks!
 

Answers and Replies

  • #2
fresh_42
Mentor
Insights Author
2021 Award
16,700
16,045
It would be easier to argue if you supply a more detailed framework. In general a representation is irreducible if there are no proper submodules. So you could take a basis vector and show that its orbit is the whole module.
 

Related Threads on Irreducibility of (anti)self-dual reps

Replies
11
Views
4K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
7
Views
28K
  • Last Post
Replies
20
Views
21K
  • Last Post
Replies
5
Views
3K
Replies
3
Views
2K
Replies
3
Views
1K
Top