nLab flag variety

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.

Related concepts

• Grassmannian

• coset space

• coadjoint orbit

• building

• Schubert variety

• Schubert calculus

• character sheaf?, horocycle correspondence, Harish Chandra transform

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

References

• 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

• Wikipedia, Generalized flag variety

• M. Brion, Lectures on the geometry of flag varieties, pdf

• Armand Borel, Linear algebraic groups, Springer

On flag varieties of loop groups:

• 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, doi:10.1016/j.physrep.2021.09.004]