Define $C,P,T$ symmetry transformation in even dimensional $d$ spacetime on a relativistic Weyl fermion theory According to https://physics.stackexchange.com/a/488388/42982 we can define $C,P,T$ discrete symmetry transformation in even dimensional spacetime.
How could we write $C,P,T$ symmetry transformation in even dimensional $d$ spacetime on a relativistic Weyl fermion? Say $d=2,4,6,8,10?$ (We consider Weyl instead of Dirac fermion because Weyl fermion is a more basic building block. Two Weyl fermions can give one Dirc fermion.)
p.s. In odd dimensional spacetime, we can define $C$, Reflection, $T$ discrete symmetry transformations. But let us focus on even dimensional spacetime here.
 A: Time reversal $T$ and parity $P$ are the two generators of the $\mathbb{Z}_2\times \mathbb{Z}_2$ in the short exact sequence $\mathbb{Z}_2\times \mathbb{Z}_2\to\mathrm{O}(p,q) \to \mathrm{SO}^+(p,q)$, i.e. they are precisely the "part" of the full orthogonal group $\mathrm{O}$ that we lose when we pass to the Lie algebra $\mathfrak{so}$/connected component of the identity $\mathrm{SO}^+$ (the "proper orthochronous transformations"). Writing down the discrete $C,P,T$ symmetries for fermions tends to be very annoying because we obtain the Dirac representation via the Clifford algebra $\mathrm{Cl}_\mathbb{C}(p,q)$, but the representation of the Pin groups (covers of the $\mathrm{O}(p,q)$ groups like the spin group for the $\mathrm{SO}(p,q)$) depends on the sign convention because unfortunately $\mathrm{Pin}(p,q)$ is not identical to $\mathrm{Pin}(q,p)$.
The Weyl spinors are projective representations of $\mathrm{SO}^+$, but not of $\mathrm{O}$: The Dirac representation of the Clifford algebra is not reducible to the Weyl representations as a representation of $\mathrm{O}$ - the action of parity on a Dirac term is precisely to "exchange" its Weyl spinor components with each other, see e.g. this answer by Nephente, so the Weyl representations are not invariant subspace with respect to parity.
A way to figure out what the correct representation of time reversal on spinors is is outlined in this answer of mine. Both charge conjugation and parity in arbitrary dimension are discussed at length in "Gamma matrices, Majorana fermions, and discrete symmetries in Minkowski and Euclidean signature" by Mike Stone.
