flag variety


Group Theory




A flag in a vector space or a projective space is a nested system of linear/projective subspaces, one of each dimension from 00 to n1n-1.

Given a field kk, the space of all flags in an nn-dimensional kk-vector space has the structure of a projective variety over kk, this is the flag variety.

If kk 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)/BSL(n,k)/B where BB is the subgroup of lower (or upper if you like) triangular matrices. By the Gram–Schmidt orthogonalization procedure, one shows the isomorphism as real manifolds SL(n,)/BSU(n)/TSL(n,\mathbb{C})/B\cong SU(n)/T where TT is the subgroup of the diagonal n×nn\times n-matrices.

More generally, the generalized flag variety is the complex projective variety obtained as the coset space G/TG /BG/T\cong G^{\mathbb{C}}/B where GG is a compact Lie group, TT its maximal torus, G G^{\mathbb{C}} the complexification of GG, which is a complex semisimple group, and BG 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. 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.

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d


Flag varieties of loop groups are discussed in

  • Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhäuser 2002

Revised on January 19, 2015 15:09:50 by Urs Schreiber (