Table 2.

The 3 × 3 matrix representation of symmetric group $S_3$ and the corresponding Möbius operators. The one-line notation on the left shows the permutations.

One-line Notation:(Permutation)	Matrix Representation (Left Action Convention)	Möbius Operator
123	$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	$\widehat{I}$
213	$\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$	$\widehat{m}$
132	$\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)$	$\widehat{M}$
321	$\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$	$\widehat{X}$
231	$\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$	$\widehat{P}$
312	$\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$	$\widehat{R}$