The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra / Michael Kapovich, Bernhard Leeb, John J. Millson.

Format:

Book

Author/Creator:

Kapovich, Michael, 1963- author.

Leeb, Bernhard, author.

Millson, John J. (John James), 1946- author.

Series:

Memoirs of the American Mathematical Society ; no. 896.

Memoirs of the American Mathematical Society, 0065-9266 ; number 896

Language:

English

Subjects (All):

Semisimple Lie groups.

Linear algebraic groups.

Geometric group theory.

Lorentz groups.

Symmetric spaces.

Rings (Algebra).

Physical Description:

1 online resource (98 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

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 $\mathbb{Q}$.

Contents:

""Contents""; ""Chapter 1. Introduction""; ""Chapter 2. Roots and Coxeter Groups""; ""2.1. Split tori over F""; ""2.2. Roots, coroots and the Langlands' dual""; ""2.3. Coxeter groups""; ""Chapter 3. The First Three Algebra Problems and the Parameter Spaces â?? for K\G/K""; ""3.1. The generalized eigenvalues of a sum problem Q1 and the parameter space â?? of k-double cosets""; ""3.2. The generalized singular values of a product and the parameter space â?? of K-double cosets""; ""3.3. The generalized invariant factor problem and the parameter space â?? of K-double cosets""

""3.4. Comparison of the parameter spaces for the four algebra problems""""3.5. Linear algebra problems""; ""Chapter 4. The existence of polygonal linkages and solutions to the algebra problems""; ""4.1. Setting up the general geometry problem""; ""4.2. Geometries modeled on Coxeter complexes""; ""4.3. Bruhat-Tits buildings associated with nonarchimedean reductive Lie groups""; ""4.4. Geodesic polygons""; ""Chapter 5. Weighted Configurations, Stability and the Relation to Polygons""; ""5.1. Gauss maps and associated dynamical systems""; ""5.2. The polyhedron D[sub(n)](X)""

""5.3. The polyhedron for the root system B[sub(2)]""""Chapter 6. Polygons in Euclidean Buildings and the Generalized Invariant Factor Problem""; ""6.1. Folding polygons into apartments""; ""6.2. A Solution of Problem Q2 is not necessarily a solution of Problem Q3""; ""Chapter 7. The Existence of Fixed Vertices in Buildings and Computation of the Saturation Factors for Reductive Groups""; ""7.1. The saturation factors associated to a root system""; ""7.2. The existence of fixed vertices""; ""7.3. Saturation factors for reductive groups""; ""Chapter 8. The Comparison of Problems Q3 and Q4""

""8.1. The Hecke ring""""8.2. A geometric interpretation of m[sub(α,β,γ)](0)""; ""8.3. The Satake transform""; ""8.4. A solution of Problem Q4 is a solution of Problem Q3""; ""8.5. A Solution of Problem Q3 is not necessarily a solution of Problem Q4""; ""8.6. The saturation theorem for GL(l)""; ""8.7. Computations for the root systems B[sub(2)] and G[sub(2)] ""; ""Appendix A. Decomposition of Tensor Products and Mumford Quotients of Products of Coadjoint orbits""; ""A. l. The existence of semistable triples and nonzero invariant vectorsin triple tensor products""

""A. 2. The semigroups of solutions to Problems Q1 and Q4""""Bibliography""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 82-83).

Description based on print version record.

ISBN:

1-4704-0502-4