  Given a finite set of roots of unity, we show that all power sums are
non-negative integers iff the set forms a group under multiplication. The main
argument is purely combinatorial and states that for an arbitrary finite set
system the non-negativity of certain alternating sums is equivalent to the set
system being a filter. As an application we determine all discrete Fourier
pairs of $\{0,1\}$-matrices. This technical result is an essential step in the
classification of $R$-matrices of quantum groups.
