Let \(G\) be a compact real Lie group, \(\rho\) a faithful irreducible representation of \(G\) in a complex vector space \(V\) and \(N\) a positive integer. The pair \((G,V)\) satisfies the \(N\)-eigenvalue property if there exists an element \(g\) of \(G\) whose conjugacy class generates \(G\) topologically and the spectrum \(X\) of \(\rho(g)\) has \(N\) elements and satisfies the no-cycle property, namely \[ u\{1,\zeta_1,\zeta_2,\dots,\zeta_{n-1}\} \nsubseteq X \] for \(2\leq n\leq N\) and all \(u \in \mathbb{C}^\ast\), where \(1,\zeta_1,\zeta_2,\dots,\zeta_{n-1}\) are the \(n\)th roots of unity. The \(N\)-eigenvalue problem is the classification of all pairs \((G,V)\) satisfying the \(N\)-eigenvalue property.
Building on earlier results for \(N = 2\) [M. H. Freedman, M. J. Larsen and Z. Wang, Commun. Math. Phys. 228, 177–199 (2002; Zbl 1045.20027); M. J. Larsen and Z. Wang, Commun. Math. Phys. 260, 641–658 (2005; Zbl 1114.57012)], motivated by applications to topological models of quantum computing and to the related representations of braid groups, the authors define the decomposability of a pair \((G,V)\) and discuss the \(N\)-eigenvalue problem in the case when \(G\) is infinite modulo its center. They first give the general shape of the solution for all \(N\) when the pair is indecomposable and then they list explicitly the indecomposable pairs which satisfy the 3-eigenvalue property. In addition, they show that a somewhat weaker condition on the eigenvalues of \(\rho(g)\) is sufficient to ensure that \(G\) is infinite modulo its center. The paper ends with two applications of the \(N\)-eigenvalue property: the first one to the Hodge-Tate theory over \(p\)-adic fields and the second one to braid group representations, in particular to braid group actions on centralizer algebras of representations of quantum groups at roots of unity which satisfy the 3-eigenvalue property.