group theory

# Contents

## 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, 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 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.

## References

Revised on March 29, 2014 03:58:13 by Urs Schreiber (185.37.147.12)