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$$\mathbb{Z}_2=\{ \mathbb{I}, -\mathbb{I} \},$$

or

$$\mathbb{Z}_2=\{ \mathbb{I}, \hat{\Pi} \},$$

where ##\hat{\Pi}## is the parity operators with the action ##\hat{\Pi}: \Psi(X) \mapsto \Psi(-X)##. My question is, which one is it? Also, say if I have the 2 component wave-function, the following also produce a representation of ##Z_2##

$$

\mathbb{Z}_2^A=\bigg\{\mathbb{I},\;\;\begin{pmatrix}

\hat{\Pi} & 0\\

0& \hat{\Pi}

\end{pmatrix} \bigg\},

$$

or

$$

\mathbb{Z}_2^B=\bigg\{\mathbb{I},\;\;\begin{pmatrix}

0 & \hat{\Pi}\\

\hat{\Pi}&0

\end{pmatrix} \bigg\}.

$$

My question is, how do you determine which one is correct when people state that the wavefunction is invariant under ##Z_2## since a wavfunction invariant under ##Z_2^A## isn't necessarily invariant under ##Z_2^B##, however they are both representations of ##Z_2##.