+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- #### Geometry +--{: .hide} [[!include higher geometry - contents]] =-- =-- =-- \tableofcontents ## Idea A __[[flag]]__ in a [[vector space]] or a [[projective space]] is a nested system of linear/projective subspaces, one of each [[dimension]] from $0$ to $n-1$. Given a [[field]] $k$, the space of all flags in an $n$-dimensional $k$-vector space has the structure of a [[projective variety]] over $k$, this is the __flag variety__. If $k$ is the field of real or complex numbers, then it has a structure of a smooth [[manifold]]. It can be considered as the [[homogeneous space]] $SL(n,k)/B$ where $B$ is the subgroup of lower (or upper if you like) triangular matrices. By the [Gram--Schmidt orthogonalization procedure](http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process), one shows the isomorphism as real manifolds $SL(n,\mathbb{C})/B\cong SU(n)/T$ where $T$ is the subgroup of the diagonal $n\times n$-matrices. 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]]. There are quantum, noncommutative and infinite-dimensional generalizations. Flag varieties have rich combinatorial and geometric structure and play an important role in [[representation theory]] and [[integrable systems]]. ## Related concepts * [[Grassmannian]] * [[coset space]] * [[coadjoint orbit]] * [[building]] * [[Schubert variety]] * [[Schubert calculus]] * [[character sheaf]], [[horocycle correspondence]], [[Harish Chandra transform]] [[!include local and global geometry - table]] ## References ### General * D. Monk, _The geometry of flag manifolds_, Proceedings of the London Mathematical Society 1959 s3-9(2):253--286; [doi:10.1112/plms/s3-9.2.253](http://dx.doi.org/10.1112/plms/s3-9.2.253) * Wikipedia, _[Generalized flag variety](http://en.wikipedia.org/wiki/Generalized_flag_variety)_ * M. Brion, _Lectures on the geometry of flag varieties_, [pdf](http://www-fourier.ujf-grenoble.fr/~mbrion/lecturesrev.pdf) * [[Armand Borel]], _Linear algebraic groups_, Springer * [[Neil Strickland]], section 5 of: *A Bestiary of Topological Objects* \[<a href="https://strickland1.org/courses/bestiary/bestiary.pdf">pdf</a>, [[Strickland-BestiaryTopological.pdf:file]]\] On flag varieties of [[loop groups]]: * {#Kumar02} [[Shrawan Kumar]], _Kac-Moody Groups, their Flag Varieties and Representation Theory_, Birkhäuser 2002 On [[flag manifold sigma-models]]: * [[Ian Affleck]], [[Dmitri Bykov]], [[Kyle Wamer]], *Flag manifold sigma models: spin chains and integrable theories*, Phys. Rept. **953** (2022) 1-93 [[arXiv:2101.11638](https://arxiv.org/abs/2101.11638), [doi:10.1016/j.physrep.2021.09.004](https://doi.org/10.1016/j.physrep.2021.09.004)] On [[rational homotopy theory|rational]] [[homotopy groups]] and [[rational cohomology]] of flag manifolds: * Haibao Duan: *Rational homotopy theory of flag manifolds* [[arXiv:2609.30591](https://arxiv.org/abs/2609.30591)] [[!include unstable classification of topological phases -- references]] [[!redirects flag varieties]] [[!redirects flag manifold]] [[!redirects flag manifolds]] [[!redirects generalized flag variety]] [[!redirects generalized flag varieties]] [[!redirects partial flag variety]] [[!redirects partial flag varieties]] [[!redirects flag space]] [[!redirects flag spaces]]