Some context first:
I'm a physics student currently taking a class in mathematical physics where we mostly talk about representation theory. We are currently talking about the irreducible representations of tensor products and therefore got some homework related to it. I'm not really familiar with the concept of tensors, I've heard of it in Linear Algebra and in Electrodynamics, but hadn't really the chance to do a lot of exercises related to it. In my homework I encountered an exercise which gives me trouble, so I hoped I could get some help here.
Problem and Definitions:
Let $V_n$ be the $(n+1)$-dimensional irreducible representation of $\mathfrak{su}(2)$, where $V_n=\operatorname{span}(\{v_j\}_{j=0}^n)$ with $v_j=F^jv_0$. Since I'm not sure if these definitions are the standard ones when dealing with the representation theory of $\mathfrak{su}(2)$ you can also refer to the wikipedia post and just change: $Y=F$ and $X=E$, the rest should be identical.
The first part of the problem was to write $V_1\otimes V_1\otimes V_1$ in terms of irreducible representations. For this I used the Clebsch-Gordan-decomposition and found $V_1\otimes V_1\otimes V_1 \cong V_3 \oplus V_1\oplus V_1$. I think this should work, at least the dimensions match, but I'm honestly not sure.
The real problem is the next part of the exercise which says that I should find the basis of the invariant subspace in $V_1\otimes V_1\otimes V_1$ equivalent to $V_3$.
What I tried so far:
I'm honestly a bit lost here. I know that since $V_3$ is a 4-dimensional representation it's basis consists of four vectors, let's say $b_i$ for $i=0,…,3$, where $b_j=F^jb_0$ and
$$Hb_0 = 3b_0,\hspace{2.0cm} Eb_0=0.$$
I also know that the basis of $V_1\otimes V_1\otimes V_1$ is something like $\{v_i\otimes v_j\otimes v_k\}_{i,j,k=0}^1$. The problem is that I don't really understand how to proceed form here. What exactly are the conditions on $v_i\otimes v_j\otimes v_k$ (and why of course) and how do they relate to the conditions for the basis of $V_3$? Another problem is that I also don't really understand how one would apply for example $F$ on $v_i\otimes v_j\otimes v_k$.
Best Answer
The vector space $V_3$ are exactly those elements that are symmetric under permutation. Thus it is easy to see that a basis is given by $$ \{ v_0\otimes v_0 \otimes v_0,\\ v_1\otimes v_1 \otimes v_1,\\ v_0\otimes v_1 \otimes v_1 + v_1\otimes v_0 \otimes v_1 + v_1\otimes v_1 \otimes v_0 ,\\ v_0\otimes v_0 \otimes v_1 + v_0\otimes v_1 \otimes v_0 + v_1\otimes v_0 \otimes v_0\} \;.$$