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In this paper we apply our results on the geometry of polygons in Cartansubspaces, symmetric spaces and buildings to four problems in algebraic grouptheory. Two of these problems are generalizations of the problems of findingthe constraints on the eigenvalues (resp. singular values) of a sum (resp.product) when the eigenvalues (singular values) of each summand (factor) arefixed. The other two problems are related to the computation of the structureconstants of the Hecke and representation rings associated with a splitreductive algebraic group over Q and its complex Langlands' dual. We give a newproof of the ``Saturation Conjecture'' for GL(m) as a consequence of oursolution of the corresponding ``saturation problem'' for the Hecke structureconstants for all split reductive algebraic groups over Q.

The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra