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In p.433 of Polchinski String theory volume 2, he said that:

In $d$ spacetime dimension,

For $d = 2 \mod 4$, each Weyl representation is its own conjugate. ($B \Gamma B^{-1} =-\Gamma $)

For $d = 0 \mod 4$, each Weyl representation is conjugate to the other. ($B \Gamma B^{-1} =+\Gamma $)

I suppose these correspond to the self or complex in his p.434 Table B 1.

However, what do these self or complex really mean?

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