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I am currently learning symmetries/group theory and I learnt that the fundamental representation and the anti-fundamental representation of $SL(2,mathbb{C})$ are not equivalent. This means that no similarity transformation can map one of them to the other.

My professor gave an explanation (on the 2nd last paragraph on page 75 of the following document http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf) but I don't see how the difference in the signs in the exponent imply that the representations are inequivalent.

Can anyone please explain the explanation of my professor, or perhaps give another explanation?

This question will probably be migrated to Math SE, but I'll post this answer anyway to clear my conscience, after @marmot helped me see the flaw in the answer I posted earlier.

$SL(2,mathbb{C})$ is the group of $2times 2$ matrices with determinant $1$. This way of defining it provides one representation using $2times 2$ matrices. To get another such representation, we can replace all of these matrices by their complex conjugates. The claim is that these two representations are inequivalent. In other words, the claim is that there is no invertible matrix $S$ that satisfies
$$
M = S^{-1}M^*S
tag{1}
$$
for all $2times 2$ matrices $M$ with $det M=1$. Equation (1) can also be written
$$
SM = M^*S.
tag{2}
$$
Now, consider matrices of the form
$$
M = left(begin{matrix} z & 0 \ 0 & 1/zend{matrix}right)
tag{3}
$$
with a complex number $zneq 0$. This has $det M=1$, and there is no non-zero matrix $S$ that satisfies (2) for all matrices of the form (3). This can be proven by writing down the most general matrix $S$ and substituting it into (2) with $M$ given by (3).