In my answer to [this question][1], there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity
$$\sum_{k=0}^{N-1}z^{2k^2+k}$$
vanish ?

The answer is clearly **No** for $N\le7$. I suspect that it never vanishes, which should answer definitively the above matricial question. Any idea of a general argument ?


  [1]: https://mathoverflow.net/q/420359