Quantum numbers of the pentaquark states ${{\rm{P}}}_{{\rm{c}}}^{+}$ via symmetry analysis

Chong-yao Chen,Muyang Chen,Yu-Xin Liu

Table 4. IRREPs of the permutation group S4 under the standard basis.

[4] p12 = p23 = p34 = p1423 = p243 = p1324 = 1, [31] ${p}_{12}=\left(\begin{array}{ccc}-1 & 0 & 0\\ -1 & 1 & 0\\ -1 & 0 & 1\end{array}\right)$, ${p}_{23}=\left(\begin{array}{ccc}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{array}\right)$, ${p}_{34}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right)$, ${p}_{1423}=\left(\begin{array}{ccc}0 & 1 & -1\\ 0 & 0 & -1\\ 1 & 0 & -1\end{array}\right)$, ${p}_{243}=\left(\begin{array}{ccc}0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{array}\right)$, ${p}_{1324}=\left(\begin{array}{ccc}0 & -1 & 1\\ 1 & -1 & 0\\ 0 & -1 & 0\end{array}\right)$, [22] ${p}_{12}=\left(\begin{array}{cc}-1 & 0\\ -1 & 1\end{array}\right)$, ${p}_{23}=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right)$, ${p}_{34}=\left(\begin{array}{cc}-1 & 0\\ -1 & 1\end{array}\right)$, ${p}_{1423}=\left(\begin{array}{cc}-1 & 0\\ -1 & 1\end{array}\right)$, ${p}_{243}=\left(\begin{array}{cc}-1 & 1\\ -1 & 0\end{array}\right)$, ${p}_{1324}=\left(\begin{array}{cc}-1 & 0\\ -1 & 1\end{array}\right)$, [211] ${p}_{12}=\left(\begin{array}{ccc}1 & 0 & 0\\ 1 & -1 & 0\\ 1 & 0 & -1\end{array}\right)$, ${p}_{23}=\left(\begin{array}{ccc}0 & -1 & 0\\ -1 & 0 & 0\\ 0 & 0 & -1\end{array}\right)$, ${p}_{34}=\left(\begin{array}{ccc}-1 & 0 & 0\\ 0 & 0 & -1\\ 0 & -1 & 0\end{array}\right)$, ${p}_{1423}=\left(\begin{array}{ccc}0 & -1 & 1\\ 0 & 0 & 1\\ -1 & 0 & 1\end{array}\right)$, ${p}_{243}=\left(\begin{array}{ccc}0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{array}\right)$, ${p}_{1324}=\left(\begin{array}{ccc}0 & 1 & -1\\ -1 & 1 & 0\\ 0 & -1 & 0\end{array}\right)$, [1111] p12 = p23 = p34 = p1423 = p1324 = − 1,p243 = 1.