Algebraic homogeneous spaces | Differential geometry

Generalized flag variety

In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of G, and they are precisely the coadjoint orbits of compact Lie groups. Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R-spaces. (Wikipedia).

Video thumbnail

What is an obtuse triangle

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Video thumbnail

What is an equilateral triangle

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Video thumbnail

Ex: Determine the Possible Number of 4 Color Striped Flags (Permutation)

This video explains how to find the number flags with 4 stripes of different colors. A permutation and the counting principle is used. Site: http://mathispower4u.com

From playlist Permutations and Combinations

Video thumbnail

What is an isosceles triangle

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Video thumbnail

What is an equiangular triangle

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Video thumbnail

Triangle Congruence (quick review)

More resources available at www.misterwootube.com

From playlist Further Properties of Geometrical Figures

Video thumbnail

What is a line bisector

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Video thumbnail

What is an acute triangle

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Video thumbnail

Using a set of points determine if the figure is a parallelogram using the midpoint formula

๐Ÿ‘‰ Learn how to determine the figure given four points. A quadrilateral is a polygon with four sides. Some of the types of quadrilaterals are: parallelogram, square, rectangle, rhombus, kite, trapezoid, etc. Each of the types of quadrilateral has its properties. Given four points that repr

From playlist Quadrilaterals on a Coordinate Plane

Video thumbnail

Flag manifolds over semifields II - Xuhua He

Workshop on Representation Theory and Geometry Topic: Flag manifolds over semifields II Speaker: Xuhua He Affiliation: Chinese University of Hong Kong; Member, School of Mathematics Date: April 03, 2021 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Eugene Gorsky - Algebra and Geometry of Link Homology 2/5

Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLY-PT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing Khovanov-Rozansky homology, focusing on connecti

From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory

Video thumbnail

Gaitsgory's central sheaves - Tom Braden

Geometric and Modular Representation Theory Seminar Topic: Gaitsgory's central sheaves Speaker: Tom Braden Affiliation: University of Massachusetts, Amherst; Member, School of Mathematics Date: February 17, 2021 For more video please visit http://video.ias.edu

From playlist Seminar on Geometric and Modular Representation Theory

Video thumbnail

Samuel Raskin: Spectral decomposition of the principal series category

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Algebraic and Complex Geometry

Video thumbnail

Singularities in reductions of Shimura varieties -Thomas Haines

Joint IAS/Princeton University Number Theory Seminar Topic: Singularities in reductions of Shimura varieties Speaker: Thomas Haines Affiliation: University of Maryland Date: May 2, 2019 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Towards a modular "2 realizations" equivalence - Simon Riche

Geometric and Modular Representation Theory Seminar Topic: Towards a modular "2 realizations" equivalence Speaker: Simon Riche Affiliation: Universitรฉ Clermont Auvergne; Member, School of Mathematics Date: May 05, 2021 For more video please visit http://video.ias.edu

From playlist Seminar on Geometric and Modular Representation Theory

Video thumbnail

Ana Caraiani - 3/3 Shimura Varieties and Modularity

We discuss vanishing theorems for the cohomology of Shimura varieties with torsion coefficients, under a genericity condition at an auxiliary prime. We describe two complementary approaches to these results, due to Caraiani-Scholze and Koshikawa, both of which rely on the geometry of the H

From playlist 2022 Summer School on the Langlands program

Video thumbnail

Induction of p-Cells and Localization - Lars Thorge Jensen

Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: Induction of p-Cells and Localization Speaker: Lars Thorge Jensen Affiliation: Member, School of Mathematics Date: November 19, 2020 For more video please visit http://video.ias.edu

From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory

Video thumbnail

Microlocal sheaves on certain affine Springer fibers - Zhiwei Yun

Geometric and Modular Representation Theory Seminar Topic: Microlocal sheaves on certain affine Springer fibers Speaker: Zhiwei Yun Affiliation: Massachusetts Institute of Technology Date: April 14, 2021 For more video please visit http://video.ias.edu

From playlist Seminar on Geometric and Modular Representation Theory

Video thumbnail

What is a scalene triangle

๐Ÿ‘‰ Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl

From playlist Types of Triangles and Their Properties

Related pages

Cohomology ring | Hopf algebra | Group representation | Lie group | Linear subspace | Smooth scheme | Principal bundle | Vector space | Linear algebra | Unitary group | Block matrix | Maximal torus | Group (mathematics) | Complex manifold | Bruhat decomposition | CW complex | Projective space | Borel subgroup | Grassmannian | Cellular homology | Classifying space | Symmetric space | Symplectic vector space | Cartan subgroup | Linear algebraic group | Projective variety | General linear group | Homomorphism | Armand Borel | Complete variety | Equivariant cohomology | Characteristic class | Hermitian symmetric space | Characteristic (algebra) | Mathematics | Field (mathematics) | Flag (linear algebra) | Primitive element (co-algebra) | Weyl group | Riemannian manifold | Sphere | Fundamental class | Exterior algebra | Cartan connection | Complexification (Lie group) | Rational point | Reductive group | Basis (linear algebra) | Compact space | Special linear group | Kรคhler manifold | Orthogonal group | Symplectic group | Parabolic Lie algebra | Homogeneous space | Maximal compact subgroup | Serre spectral sequence | Projective transformation