Memoirs of the American Mathematical Society 2008; 83 pp; softcover Volume: 192 ISBN10: 0821840541 ISBN13: 9780821840542 List Price: US$61 Individual Members: US$36.60 Institutional Members: US$48.80 Order Code: MEMO/192/896
 In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over \(\mathbb{Q}\) and its complex Langlands' dual. The authors give a new proof of the "Saturation Conjecture" for \(GL(\ell)\) as a consequence of their solution of the corresponding "saturation problem" for the Hecke structure constants for all split reductive algebraic groups over \(\mathbb{Q}\). Table of Contents  Introduction
 Roots and Coxeter groups
 The first three algebra problems and the parameter spaces \(\Sigma\) for \(K\backslash \overline{G}/K\)
 The existence of polygonal linkages and solutions to the algebra problems
 Weighted configurations, stability and the relation to polygons
 Polygons in Euclidean buildings and the generalized invariant factor problem
 The existence of fixed vertices in buildings and computation of the saturation factors for reductive groups
 The comparison of problems Q3 and Q4
 Appendix A. Decomposition of tensor products
 Appendix. Bibliography
