Irreducibility of (anti)self-dual reps

• A
• gentsagree
In summary, the second rank antisymmetric representation of the group SO(4) can be reduced into self-dual and antiself-dual representations. This can be visualized by observing that the projection of the second exterior power of the group's vector space into these subspaces commutes with the group's action. To prove that these subspaces are irreducible, a more detailed framework is needed to show that there are no proper submodules and that the orbit of a basis vector is the entire module.

gentsagree

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 $\Lambda^{2}V$ 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!

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.

What is the concept of irreducibility in (anti)self-dual reps?

Irreducibility refers to the property of a representation in which it cannot be broken down into smaller, simpler representations. In the context of (anti)self-dual reps, this means that the representation cannot be further decomposed into self-dual or anti-self-dual parts.

What is the significance of irreducibility in (anti)self-dual reps?

The concept of irreducibility is important in understanding the structure of representations and their relationship to each other. It helps identify the building blocks of representations and how they can be combined to form larger, more complex representations.

How do we determine if a representation is irreducible in (anti)self-dual reps?

To determine irreducibility in (anti)self-dual reps, we need to examine the transformation properties of the representation under certain operations. If the representation remains unchanged under these operations, then it is considered irreducible.

Can a representation be both self-dual and anti-self-dual?

No, a representation cannot be both self-dual and anti-self-dual. This is because these two properties are mutually exclusive and a representation can only have one of them.

What are the practical applications of understanding irreducibility in (anti)self-dual reps?

Understanding irreducibility in (anti)self-dual reps has applications in various fields such as physics, chemistry, and mathematics. It helps in the study of symmetry and conservation laws, as well as in the development of new mathematical models and theories.