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- Generalized_flag_variety abstract "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.The most general concept of a generalized flag variety is a conjugacy class of parabolic subgroups of a semisimple algebraic or Lie group G: G acts transitively on such a conjugacy class by conjugation, and the stabilizer of a parabolic P is P itself, so that the generalized flag variety is isomorphic to G/P. It may also be realised as the orbit of a highest weight space in a projectivized representation of G. In the algebraic setting, generalized flag varieties are precisely the homogeneous spaces for G which are complete as algebraic varieties. In the smooth setting, generalized flag manifolds are the compact flat model spaces for Cartan geometries of parabolic type, and are homogeneous Riemannian manifolds under any maximal compact subgroup of G.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.".
- Generalized_flag_variety wikiPageExternalLink books?id=hNgRLxlwL8oC.
- Generalized_flag_variety wikiPageExternalLink books?id=u3w4f63rmU8C.
- Generalized_flag_variety wikiPageExternalLink notes.html.
- Generalized_flag_variety wikiPageExternalLink sophia.pdf.
- Generalized_flag_variety wikiPageID "385339".
- Generalized_flag_variety wikiPageLength "16614".
- Generalized_flag_variety wikiPageOutDegree "89".
- Generalized_flag_variety wikiPageRevisionID "669553793".
- Generalized_flag_variety wikiPageWikiLink Armand_Borel.
- Generalized_flag_variety wikiPageWikiLink Basis_(linear_algebra).
- Generalized_flag_variety wikiPageWikiLink Block_matrix.
- Generalized_flag_variety wikiPageWikiLink Borel_subgroup.
- Generalized_flag_variety wikiPageWikiLink CW_complex.
- Generalized_flag_variety wikiPageWikiLink Cartan_subgroup.
- Generalized_flag_variety wikiPageWikiLink Category:Algebraic_homogeneous_spaces.
- Generalized_flag_variety wikiPageWikiLink Category:Differential_geometry.
- Generalized_flag_variety wikiPageWikiLink Cellular_homology.
- Generalized_flag_variety wikiPageWikiLink Characteristic_class.
- Generalized_flag_variety wikiPageWikiLink Classical_Lie_group.
- Generalized_flag_variety wikiPageWikiLink Classical_group.
- Generalized_flag_variety wikiPageWikiLink Classifying_space.
- Generalized_flag_variety wikiPageWikiLink Cohomology_ring.
- Generalized_flag_variety wikiPageWikiLink Compact_space.
- Generalized_flag_variety wikiPageWikiLink Complete_variety.
- Generalized_flag_variety wikiPageWikiLink Complex_manifold.
- Generalized_flag_variety wikiPageWikiLink Complexification_(Lie_group).
- Generalized_flag_variety wikiPageWikiLink Conformal_map.
- Generalized_flag_variety wikiPageWikiLink Conformal_transformation.
- Generalized_flag_variety wikiPageWikiLink Conjugacy_class.
- Generalized_flag_variety wikiPageWikiLink Differentiable_manifold.
- Generalized_flag_variety wikiPageWikiLink Dimension_(linear_algebra).
- Generalized_flag_variety wikiPageWikiLink Dimension_(vector_space).
- Generalized_flag_variety wikiPageWikiLink Elementary_symmetric_polynomial.
- Generalized_flag_variety wikiPageWikiLink Elementary_symmetric_polynomials.
- Generalized_flag_variety wikiPageWikiLink Equivariant_cohomology.
- Generalized_flag_variety wikiPageWikiLink Exterior_algebra.
- Generalized_flag_variety wikiPageWikiLink Field_(mathematics).
- Generalized_flag_variety wikiPageWikiLink Filtration_(abstract_algebra).
- Generalized_flag_variety wikiPageWikiLink Filtration_(mathematics).
- Generalized_flag_variety wikiPageWikiLink Flag_(linear_algebra).
- Generalized_flag_variety wikiPageWikiLink Fundamental_class.
- Generalized_flag_variety wikiPageWikiLink General_linear_group.
- Generalized_flag_variety wikiPageWikiLink Grassmannian.
- Generalized_flag_variety wikiPageWikiLink Group_(mathematics).
- Generalized_flag_variety wikiPageWikiLink Group_action.
- Generalized_flag_variety wikiPageWikiLink Group_representation.
- Generalized_flag_variety wikiPageWikiLink Hermitian_symmetric_space.
- Generalized_flag_variety wikiPageWikiLink Highest_weight.
- Generalized_flag_variety wikiPageWikiLink Homogeneous_space.
- Generalized_flag_variety wikiPageWikiLink Homography.
- Generalized_flag_variety wikiPageWikiLink Homomorphism.
- Generalized_flag_variety wikiPageWikiLink Hopf_algebra.
- Generalized_flag_variety wikiPageWikiLink Indefinite_orthogonal_group.
- Generalized_flag_variety wikiPageWikiLink Inner_product.
- Generalized_flag_variety wikiPageWikiLink Inner_product_space.
- Generalized_flag_variety wikiPageWikiLink Isotropic.
- Generalized_flag_variety wikiPageWikiLink Isotropy.
- Generalized_flag_variety wikiPageWikiLink Kähler_manifold.
- Generalized_flag_variety wikiPageWikiLink Lie_group.
- Generalized_flag_variety wikiPageWikiLink Linear_algebra.
- Generalized_flag_variety wikiPageWikiLink Linear_algebraic_group.
- Generalized_flag_variety wikiPageWikiLink Linear_subspace.
- Generalized_flag_variety wikiPageWikiLink Mathematics.
- Generalized_flag_variety wikiPageWikiLink Maximal_torus.
- Generalized_flag_variety wikiPageWikiLink Orthogonal_group.
- Generalized_flag_variety wikiPageWikiLink Orthonormal.
- Generalized_flag_variety wikiPageWikiLink Orthonormality.
- Generalized_flag_variety wikiPageWikiLink Parabolic_Lie_algebra.
- Generalized_flag_variety wikiPageWikiLink Parabolic_geometry.
- Generalized_flag_variety wikiPageWikiLink Parabolic_subgroup.
- Generalized_flag_variety wikiPageWikiLink Primitive_element_(co-algebra).
- Generalized_flag_variety wikiPageWikiLink Principal_bundle.
- Generalized_flag_variety wikiPageWikiLink Projective_algebraic_variety.
- Generalized_flag_variety wikiPageWikiLink Projective_space.
- Generalized_flag_variety wikiPageWikiLink Projective_transformation.
- Generalized_flag_variety wikiPageWikiLink Projective_variety.
- Generalized_flag_variety wikiPageWikiLink Real_form.
- Generalized_flag_variety wikiPageWikiLink Real_form_(Lie_theory).
- Generalized_flag_variety wikiPageWikiLink Riemannian_manifold.
- Generalized_flag_variety wikiPageWikiLink Semisimple_algebraic_group.
- Generalized_flag_variety wikiPageWikiLink Semisimple_group.
- Generalized_flag_variety wikiPageWikiLink Serre_spectral_sequence.
- Generalized_flag_variety wikiPageWikiLink Smooth_manifold.
- Generalized_flag_variety wikiPageWikiLink Special_linear_group.
- Generalized_flag_variety wikiPageWikiLink Sphere.
- Generalized_flag_variety wikiPageWikiLink Split_orthogonal_group.
- Generalized_flag_variety wikiPageWikiLink Stabilizer_(group_theory).
- Generalized_flag_variety wikiPageWikiLink Symmetric_space.
- Generalized_flag_variety wikiPageWikiLink Symplectic_group.
- Generalized_flag_variety wikiPageWikiLink Symplectic_vector_space.
- Generalized_flag_variety wikiPageWikiLink Triangular_matrix.
- Generalized_flag_variety wikiPageWikiLink Unitary_group.
- Generalized_flag_variety wikiPageWikiLink Upper_triangular_matrix.
- Generalized_flag_variety wikiPageWikiLink Vector_space.
- Generalized_flag_variety wikiPageWikiLink Weight_(representation_theory).
- Generalized_flag_variety wikiPageWikiLink Weight_space.
- Generalized_flag_variety wikiPageWikiLink Weyl_group.
- Generalized_flag_variety wikiPageWikiLinkText "Flag varieties".
- Generalized_flag_variety wikiPageWikiLinkText "Generalized flag variety".