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What is meaning concretely than an expression is $Z_2$ symmetric under a certain transformation ?

Does is mean that the expression is the same when we do the transformation ? Does is mean that the expression is the opposite when we do the transformation ?

For the context : I have in mind the expression of a Beyond Standard Model Higgs potential :

It is said in an article than a $Z_2$ discrete symmetry is applied the potential (a function of $\phi_1$ and $\phi_2$) : "a $Z_2$ discrete symmetry of the form $\phi_1\rightarrow -\phi_1$ is imposed on the potential".

https://www.itp.kit.edu/_media/publications/masterthesismarcel.pdf page 8

Mathieu Krisztian
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    $\Bbb Z_2$ acts on whatever it makes sense to act on by letting $1$ be the identity map and $-1$ being a sign flip. I am yet to see a situation where $\Bbb Z_2$-symmetry means anything else than your equation being invariant under a sign-flip in the unknown. – Ivo Terek Jan 02 '21 at 19:02
  • @Ivo Terek : thank you – Mathieu Krisztian Jan 02 '21 at 19:40

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