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The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra
Michael Kapovich, University of California, Davis, CA, Bernhard Leeb, Universität München, Munich, Germany, and John J. Millson, University of Maryland, College Park, MD
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Memoirs of the American Mathematical Society
2008; 83 pp; softcover
Volume: 192
ISBN-10: 0-8218-4054-1
ISBN-13: 978-0-8218-4054-2
List Price: US$65 Individual Members: US$39
Institutional Members: US\$52
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 $$\mathbb{Q}$$.

• 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