Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book discusses the importance of flag varieties in geometric objects and elucidates its richness as interplay of geometry, combinatorics and representation theory. The book presents a discussion on the representation theory of complex semisimple Lie algebras, as well as the representation theory of semisimple algebraic groups. In addition, the book also discusses the representation theory of symmetric groups. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Many of the geometric results admit elegant combinatorial description because of the root system connections, a typical example being the description of the singular locus of a Schubert variety. This discussion is carried out as a consequence of standard monomial theory. Consequently, this book includes standard monomial theory and some important applications—singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. The two recent results on Schubert varieties in the Grassmannian have also been included in this book. The first result gives a free resolution of certain Schubert singularities. The second result is about certain Levi subgroup actions on Schubert varieties in the Grassmannian and derives some interesting geometric and representation-theoretic consequences.

Mathematics. --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Group theory. --- Group Theory and Generalizations. --- Associative Rings and Algebras. --- Algebraic Geometry. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Algebraic geometry --- Geometry --- Math --- Science --- Geometry, Algebraic. --- Flag manifolds. --- Representations of groups. --- Semisimple Lie groups. --- Schubert varieties. --- MATHEMATICS -- Geometry -- General. --- Geometry, Algebraic --- Semi-simple Lie groups --- Lie groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Flag varieties (Mathematics) --- Manifolds, Flag --- Varieties, Flag (Mathematics) --- Algebraic varieties

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Flag manifolds. --- Affine algebraic groups. --- Quantum groups. --- Schur complement. --- Kazhdan-Lusztig polynomials. --- Algebra, Homological. --- Variétés de drapeaux (géométrie) --- Groupes algébriques affines. --- Groupes quantiques. --- Complément de Schur. --- Algèbre homologique.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Maximal functions. --- Littlewood-Paley theory. --- Fourier transformations. --- Riesz spaces. --- Flag manifolds. --- Hardy spaces. --- Functions of several real variables. --- Functions of complex variables.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This monograph, divided into four parts, presents a comprehensive treatment and systematic examination of cycle spaces of flag domains. Assuming only a basic familiarity with the concepts of Lie theory and geometry, this work presents a complete structure theory for these cycle spaces, as well as their applications to harmonic analysis and algebraic geometry. Key features: * Accessible to readers from a wide range of fields, with all the necessary background material provided for the nonspecialist * Many new results presented for the first time * Driven by numerous examples * The exposition is presented from the complex geometric viewpoint, but the methods, applications and much of the motivation also come from real and complex algebraic groups and their representations, as well as other areas of geometry * Comparisons with classical Barlet cycle spaces are given * Good bibliography and index Researchers and graduate students in differential geometry, complex analysis, harmonic analysis, representation theory, transformation groups, algebraic geometry, and areas of global geometric analysis will benefit from this work.

Semisimple Lie groups. --- Flag manifolds. --- Twistor theory. --- Automorphic forms. --- Homogeneous spaces. --- Spaces, Homogeneous --- Lie groups --- Automorphic functions --- Forms (Mathematics) --- Twistors --- Congruences (Geometry) --- Field theory (Physics) --- Space and time --- Flag varieties (Mathematics) --- Manifolds, Flag --- Varieties, Flag (Mathematics) --- Algebraic varieties --- Semi-simple Lie groups --- Global differential geometry. --- Topological Groups. --- Differential equations, partial. --- Global analysis. --- Geometry, algebraic. --- Quantum theory. --- Differential Geometry. --- Topological Groups, Lie Groups. --- Several Complex Variables and Analytic Spaces. --- Global Analysis and Analysis on Manifolds. --- Algebraic Geometry. --- Quantum Physics. --- Global analysis (Mathematics) --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Algebraic geometry --- Geometry --- Partial differential equations --- Groups, Topological --- Continuous groups --- Geometry, Differential --- Differential geometry. --- Topological groups. --- Lie groups. --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Algebraic geometry. --- Quantum physics. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Topology --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential geometry

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The story of the five Marines and one Navy corpsman that were forever immortalized as a symbol of WWII by raising the American flag at the battle of Iwo Jima. When Joe Rosenthal's photograph of the event becomes a symbol of hope for the families at home, the three surviving men are pulled from combat and sent on a tour across America to raise desperately-needed bond money. It is a trip that brings out the truths of both that symbolic act, and of their lives during war

Iwo Jima, Battle of, Japan, 1945 --- Photographs --- Marines --- World War, 1939-1945 --- Flag raising and lowering --- War photography --- Bradley, James, - 1954 --- -Powers, Ron --- Bradley, John, - 1923-1994 --- Hayes, Ira, - 1923-1955 --- Gagnon, Rene --- Iwo Jima (Volcano Islands, Japan)