A theorem on roots of unity and a combinatorial principle

Abstract

Given a finite set of roots of unity, we show that all power sums are nonnegative 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.

Factorizable R-Matrices for Small Quantum Groups

“…The necessity of this form (in particular that the support of f is indeed a subgroup!) amounts to a combinatorial problem of its own interest, which we solved for π 1 cyclic in [10] and for Z 2 ×Z 2 by hand; a closed proof for all abelian groups would be interesting. Definition 3.4.…”

New R-matrices for Small Quantum Groups

“…More precisely, the extensions u q (g, Λ) of u q (g) we consider depend on a choice of a lattice Λ R ⊂ Λ ⊂ Λ W between root and weight lattice, which corresponds to a choice of a complex connected Lie group associated to g. We first derive a necessary form of the R-matrix, depending only on the fundamental group Λ W /Λ R ; this amounts to a question in additive combinatorics we have settled in [LN14]. The main calculations concluding the present article is to check sufficiency in terms of certain sublattices of Λ.…”

“…We will give all solutions of the group-equations of a group G, where G is cyclic or equal to Z 2 × Z 2 , since these are the relevant cases for G = π 1 the fundamental group of the Lie algebras in interest. The case g = A n with fundamental group Z n+1 is particularly hard and depends on a question in additive combinatorics, which we settled in [LN14].…”