Let $u$ and $v$ be two vectors in $\mathbb{C}^n$.
Define a permutation of a vector $v':=\sigma(v)$ by $v'\_j = v\_{\sigma(j)}$ for any $\sigma \in S\_n$.


It is easy to show the following for $u,v \in \mathbb{C}^n$ by considering $\sigma$ being a swap:



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> $u = c\vec{1}$ or $v = c\vec{1}$ for some $c \in \mathbb{C} \iff $ $\langle u,\sigma(v) \rangle$ is invariant for any $\sigma \in S\_n$.
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I am wondering if the following is also true:



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> Let $n \geq 3$. $u = c\vec{1}$ or $v = c\vec{1}$ for some $c \in \mathbb{C} \iff $ $\left|\langle u,\sigma(v) \rangle \right|$ is invariant for any $\sigma \in S\_n$.
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Note that for $n=2$ there is a simple counterexample $u=v=(1,-1)$. However I was unable to find any counterexample for $n \geq 3$.


**Update.** The conjecture is false in light of Matt's answer. However the known counterexamples are all highly "structural". Is there some way to identify all counterexamples?



Use random inner product to test if at least one vector is uniform
This (and specifically the $\Leftarrow$) is false: let
$$u = (-1,0,1)$$
$$v = (1, \omega, \omega^2)$$
where $\omega$ is a cube root of unity.


