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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 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 $\frac12 \otimes \frac12 \otimes 1$?

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.

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 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 $\frac12 \otimes \frac12 \otimes 1$?

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 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 $\frac12 \otimes \frac12 \otimes 1$?