On Nonzero Kronecker Coefficients and their Consequences for Spectra

A triple of spectra (r, r , r) is said to be admissible if there is a density operator ρ with (Spec ρ, Spec ρ, Spec ρ) = (r, r , r). How can we characterise such triples? It turns out that the admissible spectral triples correspond to Young diagrams (μ, ν, λ) with nonzero Kronecker coefficient gμνλ [4, 13]. This means that the irreducible representation Vλ is contained in the tensor product of Vμ and Vν . Here, we show that such triples form a finitely generated semigroup, thereby resolving a conjecture of Klyachko [13]. As a consequence we are able to obtain stronger results than in [4] and give a complete information-theoretic proof of the correspondence between triples of spectra and representations. Finally, we show that spectral triples form a convex polytope.