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.


Revised on March 29, 2014 03:58:13 by Urs Schreiber (