$\ell_1$ norm is tagged as non-convex by Julia I have an optimization problem where the objective function is
$$\text{minimize}_{P \in S_n} ~~ \| PXP'-Y \|_1$$
over the set of permutation matrices $S_n$. However, my solver in Julia (convex package) says the objective function is not DCP.
Since it is the $\ell_1$ norm, I expect it to be convex.  Can anyone explain me why it is not convex?

$f(g(\cdot))$ is not convex just because $f(\cdot)$ is convex. A sufficient condition is, e.g., $f(\cdot)$ convex non-decreasing and $g(\cdot)$ convex.
In your case, simply plot the scalar function $|p^2-1|$ by hand and convince yourself that it is nonconvex.
(skipping the general problem that the set of permutation matrices is nonconvex to begin with)
