How exactly is the EM field tensor related to the proper orthochronous Lorentz group?
What do you mean by "related". As a rank 2 tensor, the Faraday tensor transforms under a proper orthochronous Lorentz transformation just like any other rank 2 tensor:
I mean the specific representation of the group, why is it the direct sum of the (1,0) and (0,1) irreps, is it related with the tensor being a 2-form and its parity invariance?
It has to do with the parity invariance. Just like in the Dirac field case, it's the reason why we need the direct sum of elementary irreducible representations.
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