Spinor Representation of $O(3)$ for $S_4$

Abstract

All possible permutations in the discrete $S_4$ group are classified by three rotation angles associated with the orthogonal group $O(3)$. We construct a spinor representation ${\bf 2}_D$ of $O(3)$, which is transformed by three 4$\times$4 matrices corresponding to three Pauli matrices in $SO(3)$. An irreducible decomposition of ${\bf 2}_D \otimes {\bf 2}_D$ supplies a vector representation of {\bf 3} of $O(3)$, thereby, of $S_4$. Our construction is consistent with the mathematical fact that $O(3)=SO(3)\times \boldsymbol{Z}_2$. The $\boldsymbol{Z}_2$ parity in the spinorial space is described by a block off-diagonal matrix as the spinorial parity operator, whose eigenvalues are $\pm 1$ consistent with $\boldsymbol{Z}_2$.