# The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions

Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions)

My question is that what does the conjugacy mean here precisely? What do the mutual and the self mean in terms of the conjugacy?

Let the spacetime metric have signature $$(s,t)$$.
1. Mutual conjugated means that the left and right Weyl representations are each other's complex conjugate complex representations. This happens if $$s-t~=~2\mod 4$$.
2. Self-conjugated means that the left and right Weyl representations are (pseudo) real representations. This happens if $$s-t~=~0\mod 4$$.
(Weyl representations only exist in even spacetime dimensions $$s+t~=~0\mod 2$$.)