Representations of SU(3) Algebra

[dman12]

Homework Statement

I'm trying to figure out this question:

"Show that the 10-dimensional representation R_3,0 of A₂ corresponds to a reducible representation of the L_C[SU(2)] subalgebra corresponding to any root. Find the irreducible components of this representation. Does the answer depend on the particular root chosen?"

Homework Equations

The Attempt at a Solution

So I am happy finding the decuplet of A₂, which is just complexified L[SU(3)]. You get a triangle of the 10 weights, which are two dimensional due to the fact that the Lie algebra is rank 2.

But I don't get how you can view this as being a reducible representation of complexified L[SU(2)]. The weights of L[SU(2)] representations are one dimensional so how can we build a two dimensional weight system from them?

Any guidance on how to go about this question would be much appreciated!

 

[mark726]

Thank you for your question. I understand your confusion regarding the representation of A2 and its relationship to the LC[SU(2)] subalgebra. Let me try to explain it in a simpler way.

Firstly, it is important to note that the representation R3,0 of A2 is a 10-dimensional representation, meaning it has 10 different states or "weights". These weights are actually 2-dimensional, not 1-dimensional as you mentioned. This is because the representation is 10-dimensional, not 10 copies of a 1-dimensional representation.

Now, the LC[SU(2)] subalgebra corresponds to a root in A2. This root can be represented by a vector in 2-dimensional space. For example, let's say the root is (1,0), then this corresponds to the LC[SU(2)] subalgebra that contains the elements [Jz, J+] and [Jz, J-]. Similarly, if the root is (0,1), then the subalgebra contains [Jx, J+] and [Jx, J-].

The key point here is that the weights of the representation R3,0 can be expressed as a linear combination of these two subalgebras. This means that the representation R3,0 is actually a reducible representation, as it can be broken down into smaller, irreducible representations of the LC[SU(2)] subalgebras.

To find the irreducible components of this representation, you can use a technique called "weight diagrams". This involves drawing a diagram with the weights as points and connecting them with lines to show how they can be decomposed into smaller representations. The answer to whether this depends on the particular root chosen is yes, as the weight diagram will be different for different roots.

I hope this helps to clarify the concept for you. Please let me know if you have any further questions or need more guidance. Good luck with your studies!