# Operators commuting with tensor product representations of SU(2)

I am currently investigating $$SU(2)$$ symmetric qubit systems. In the course of this work I proved the following theorem:

Let $$S_n$$ denote the permutation group of $$n$$ elements. For $$\sigma\in S_n$$ define the tensor product permutation operator $$P_\sigma\in \operatorname{End}((\mathbb{C^2})^{\otimes n})$$ as $$P_\sigma: v_1 \otimes \cdots \otimes v_n \mapsto v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(n)}$$

Let $$T \in \operatorname{End}((\mathbb{C^2})^{\otimes n})$$ be a linear operator. Then

$$\left(\exists \{c_\sigma\}_{\sigma \in S_n} \subset \mathbb{C}: T=\sum_{\sigma\in S_n} c_\sigma P_\sigma\right)\iff \forall U \in SU(2): [T, U^{\otimes n}]=0$$

In words: $$T$$ commutes with the tensor product of the fundamental representation of $$SU(2)$$ if and only if $$T$$ is a linear combination of permutation operators.

The implication “$$\implies$$” is trivially true. However I also proved the other direction. Now my question is whether this theorem is already known? If yes, I would be grateful if someone could point me to some references.

## 1 Answer

Congrats! You have rediscovered on your own a cornerstone of representation theory. This is the First Fundamental Theorem of Classical Invariant Theory for the group $$SU(2)$$ or $$SL(2)$$. The easy part $$\Rightarrow$$ was discovered by Arthur Cayley in:

• A. Cayley, On linear transformations, Cambridge Dublin Math. J. 1 (1846) 104–122.

The hard part $$\Leftarrow$$ was first proved by Alfred Clebsch in:

• A. Clebsch, Ueber symbolische Darstellung algebraischer Formen, J. Reine Angew. Math. 59 (1861) 1–62.

You can find a proof in my two MathOverflow answers:

https://mathoverflow.net/questions/255492/how-to-constructively-combinatorially-prove-schur-weyl-duality/255853#255853

and

https://mathoverflow.net/questions/255492/how-to-constructively-combinatorially-prove-schur-weyl-duality/255938#255938