Table 1.

The cyclic permutation basis expanded in the computational basis. $ħ=1$ is considered

$|s,m⟩$	Expansion in ${\{|\uparrow ⟩,|\downarrow ⟩\}}^{\otimes 3}$	${\widehat{P}}_{+}$’s Eigenvalue
$|A,3/2⟩$	$|\uparrow \uparrow \uparrow ⟩$	1
$|A,1/2⟩$	$\frac{1}{\sqrt{3}}\left(|\uparrow \uparrow \downarrow ⟩+|\downarrow \uparrow \uparrow ⟩+|\uparrow \downarrow \uparrow ⟩\right)$	1
$|A,-1/2⟩$	$\frac{1}{\sqrt{3}}\left(|\downarrow \downarrow \uparrow ⟩+|\uparrow \downarrow \downarrow ⟩+|\downarrow \uparrow \downarrow ⟩\right)$	1
$|A,-3/2⟩$	$|\downarrow \downarrow \downarrow ⟩$	1
$|E_{+},1/2⟩$	$\frac{1}{\sqrt{3}}\left(|\uparrow \uparrow \downarrow ⟩+{\mathit{\epsilon }}^{\ast }|\downarrow \uparrow \uparrow ⟩+\mathit{\epsilon }|\uparrow \downarrow \uparrow ⟩\right)$	$\mathit{\epsilon }$
$|E_{+},-1/2⟩$	$\frac{1}{\sqrt{3}}\left(|\downarrow \downarrow \uparrow ⟩+{\mathit{\epsilon }}^{\ast }|\uparrow \downarrow \downarrow ⟩+\mathit{\epsilon }|\downarrow \uparrow \downarrow ⟩\right)$	$\mathit{\epsilon }$
$|E_{-},1/2⟩$	$\frac{1}{\sqrt{3}}\left(|\uparrow \uparrow \downarrow ⟩+\mathit{\epsilon }|\downarrow \uparrow \uparrow ⟩+{\mathit{\epsilon }}^{\ast }|\uparrow \downarrow \uparrow ⟩\right)$	${\mathit{\epsilon }}^{\ast }$
$|E_{-},-1/2⟩$	$\frac{1}{\sqrt{3}}\left(|\downarrow \downarrow \uparrow ⟩+\mathit{\epsilon }|\uparrow \downarrow \downarrow ⟩+{\mathit{\epsilon }}^{\ast }|\downarrow \uparrow \downarrow ⟩\right)$	${\mathit{\epsilon }}^{\ast }$