Alt Tensor: If Alt(\omega)=\omega, Is \omega Alternating?

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If \omega is an alternating tensor, then Alt(\omega)=\omega, where Alt is the mapping that maps any tensor to an alternating tensor.

I guess the converse is also true, i.e., if Alt(\omega)=\omega, then \omega must be an alternating tensor. Am I right?
 
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If you have w=Alt(w), and Alt(w) is an alternating tensor...
 

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