Operators commuting with tensor product representations of SU(2)

Question

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.

Answer 1

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