Generalized triangle inequalities and their applications

We present a survey of our recent work on generalized triangle inequalities in infinitesimal symmetric spaces, nonpositively curved symmetric spaces and Euclidean buildings. We also explain how these results can be used to analyze some basic problems of algebraic group theory including the problem of decomposition of tensor products of irreducible representations of complex reductive Lie groups. Among the applications is a generalization of the Saturation Theorem of Knutson and Tao to Lie groups other than SL(n,C). Mathematics Subject Classification (2000). Primary 22E46, 20G15, 53C20; Secondary 14L24, 20E42.