While studying the representation theory, I came up with the following example, but it seems hard for me to solve.
For an integer or half-integer $j$, let $V_j$ be a $(2j+1)$-dimensional complex vector space with basis $$\{{|j, j\rangle, |j, j-1\rangle, \cdots, |j,-j\rangle}\}$$ which transforms as an irreducible representation of $SU(2)$. Consider the tensor product $V_j\otimes V_j$. Generally this space is reducible, and it is well known that $$V_j\otimes V_j \approx V_0 \oplus V_1 \oplus \cdots \oplus V_{2j}$$ as a direct sum of irreducible representation. Then, what is the explicit element in $V_0$ on RHS in terms of the tensor product state?
- This question is motivated from my pure interest. This is not the homework-like question.
