Flag manifold

In mathematics , a flag manifold is the space of complete flags in a vector space or, more generally, the quotient of a semi-simple algebraic group after a Borel subgroup . Flag manifolds are projective varieties .

Flag manifold of a vector space

A complete flag in a finite-dimensional (real or complex) vector space is a sequence ${\displaystyle V}$

${\displaystyle (V_{0},V_{1},\ldots ,V_{n})}$

of subspaces of with and , so that each subspace is properly contained in the following, i. H. ${\displaystyle V}$ ${\displaystyle V_{0}=0}$ ${\displaystyle V_{n}=V}$

${\displaystyle V_{0}\subsetneq V_{1}\subsetneq \ldots \subsetneq V_{n}}$

and so that

${\displaystyle \dim V_{i}=i}$

for applies, especially so . ${\displaystyle i=0,\ldots ,n}$ ${\displaystyle n=\dim(V)}$

The general linear group acts transitively on the set of all complete flags, the stabilizers of a flag are conjugate to the group of invertible upper triangular matrices . So there is a bijection between and the set of all complete flags. Therefore will ${\displaystyle \mathrm {GL} (V)}$ ${\displaystyle B}$ ${\displaystyle GL(V)/B}$

${\displaystyle {\mathcal {F}}l(V):=GL(V)/B}$

referred to as the flag manifold .

The canonical embedding in the product of Graßmann manifolds

${\displaystyle {\mathcal {F}}l(V)\subset Gr(1,n)\times Gr(2,n)\times \ldots \times Gr(n,n)}$

makes the flag manifold a differentiable manifold and (by means of the Plücker embedding of the Graßmann manifolds) a projective variety.

Generalized flag manifolds

Let it be a semi-simple Lie group and a Borel group , i. H. a minimal parabolic subset of . Then the homogeneous space is called a generalized plume manifold . If is an algebraic group, is a projective variety. ${\displaystyle G}$ ${\displaystyle B\subset G}$ ${\displaystyle G}$ ${\displaystyle G/B}$ ${\displaystyle G}$ ${\displaystyle G/B}$

The above examples of the flag manifolds of a vector space are obtained for or and the subgroup of the invertible upper triangular matrices. ${\displaystyle G=GL(n,\mathbb {R} )}$ ${\displaystyle G=GL(n,\mathbb {C} )}$ ${\displaystyle B}$

literature

• Charles Ehresmann : Sur la topologie de certains espaces homogènes. Ann. of Math. (2) 35 (1934), no. 2, 396-443.
• Shiing-Shen Chern : On the characteristic classes of complex sphere bundles and algebraic varieties. Amer. J. Math. 75, (1953). 565-597.
• Armand Borel : Cohomologie des espaces homogènes. Séminaire Bourbaki, Vol. 1, Exp. 45, 371-378, Soc. Math. France, Paris, 1995.
• DV Alekseevsky: Flag manifolds. 11th Yugoslav Geometrical Seminar (Divčibare, 1996). E.g. Rad. Mat. Inst. Beograd. (NS) 6 (14): 3-35 (1997). online (PDF)

Web links

• Flag Manifold (MathWorld)
• Flag Variety (nLab)
• What are flag manifolds and why are they interesting? (Australian Mathematical Society Web Site - the Gazette)
• Brion: Lectures on the geometry of flag varieties