A theorem on roots of unity and a combinatorial principle

Simon Lentner; Daniel Nett

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arXiv · 1409.5822

A theorem on roots of unity and a combinatorial principle

Simon Lentner·Daniel Nett

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Abstract

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.

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Simon Lentner, Daniel Nett. 2014-09-19. A theorem on roots of unity and a combinatorial principle. https://arxiv.org/abs/1409.5822

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