Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThat's not well put at [[flag variety]], is it? >More generally, the __generalized flag variety__ is the complex projective variety obtained as the [[coset space]] $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a [[compact Lie group]], $T$ its [[maximal torus]], $G^{\mathbb{C}}$ the [[complexification]] of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ is the [[Borel subgroup]]. It has a structure of a compact [[Kähler manifold]]. It is a special case of the larger family of coset spaces of semisimple groups modulo [[parabolic subgroup|parabolics]] which includes, for example, [[Grassmannians]]. The 'larger family' are the generalized flag varieties, no?

   That’s not well put at flag variety, is it?

   More generally, the generalized flag variety is the complex projective variety obtained as the coset space $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a compact Lie group, $T$ its maximal torus, $G^{\mathbb{C}}$ the complexification of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ is the Borel subgroup. It has a structure of a compact Kähler manifold. It is a special case of the larger family of coset spaces of semisimple groups modulo parabolics which includes, for example, Grassmannians.

   The ’larger family’ are the generalized flag varieties, no?

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeDec 19th 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItexDepends on the author. You first have the usual falg variety GL(n)/B then you generalize to generalized G/B and finally to even more generalized G/P. Now the generalized ones are either the middle ones or the most general ones in the list, depending on the author.

   Depends on the author. You first have the usual falg variety GL(n)/B then you generalize to generalized G/B and finally to even more generalized G/P. Now the generalized ones are either the middle ones or the most general ones in the list, depending on the author.