The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra
Author(s)
Bibliographic Information
The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra
(Memoirs of the American Mathematical Society, no. 896)
American Mathematical Society, 2008
Available at 11 libraries
  Aomori
  Iwate
  Miyagi
  Akita
  Yamagata
  Fukushima
  Ibaraki
  Tochigi
  Gunma
  Saitama
  Chiba
  Tokyo
  Kanagawa
  Niigata
  Toyama
  Ishikawa
  Fukui
  Yamanashi
  Nagano
  Gifu
  Shizuoka
  Aichi
  Mie
  Shiga
  Kyoto
  Osaka
  Hyogo
  Nara
  Wakayama
  Tottori
  Shimane
  Okayama
  Hiroshima
  Yamaguchi
  Tokushima
  Kagawa
  Ehime
  Kochi
  Fukuoka
  Saga
  Nagasaki
  Kumamoto
  Oita
  Miyazaki
  Kagoshima
  Okinawa
  Korea
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Note
Bibliography: p. 82-83
Description and Table of Contents
Description
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.
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