# Are Pauli matrices invariant tensors in the representation of $\frac12 \otimes \frac12 \otimes 1$?

If we raise the index of the Pauli matrices with Levi-Civita symbol $$\epsilon$$ we obtain the 2-index spinors $$(\sigma_i)^{AB} = (\sigma_i)^A{}_C \ \epsilon^{CB}$$. The textbook (Ref. 1) argued that they are invariant tensors in the representation $$\frac12 \otimes \frac12 \otimes 1$$. However, I raised the index by $$\epsilon$$, which gives that: $$(\sigma_1)^{AB}=\left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right), \quad (\sigma_2)^{AB}=\left( \begin{array}{cc} i & 0 \\ 0 & i \\ \end{array} \right),\quad (\sigma_3)^{AB}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).$$ How can I verify they are invariant under the representation $$\frac12 \otimes \frac12 \otimes 1$$?

Reference:

1. Carlo Rovelli and Francesca Vidotto, Covariant Loop Quantum Gravity, Exercise 1.9 pp. 27. ISBN:9781107069626.

Yes, the Pauli matrices are invariant in the sense that $$\sum_{j=1}^3 U \sigma_j U^{-1}(R^{-1})^j{}_k~=~ \sigma_k ,\tag{A}$$ where $$U\in SU(2)$$ is a $$2\times 2$$-matrix in the spin-$$1/2$$ representation and $$R\in SO(3)$$ is the corresponding $$3\times 3$$-matrix in the spin-$$1$$ representation, cf. e.g. my Phys.SE answer here.