Editing Generalized flag variety

In [[mathematics]], a '''generalized flag variety''' (or simply '''flag variety''') is a [[homogeneous space]] whose points are [[flag (linear algebra)|flags]] in a finite-dimensional [[vector space]] ''V'' over a [[field (mathematics)|field]] '''F'''. When '''F''' is the real or complex numbers, a generalized flag variety is a [[smooth manifold|smooth]] or [[complex manifold]], called a '''real''' or '''complex''' '''flag manifold'''. Flag varieties are naturally [[projective variety|projective varieties]].

Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field '''F''', which is a flag variety for the [[special linear group]] over '''F'''. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the [[symplectic group]]. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags.

In the most general sense, a generalized flag variety is defined to mean a '''projective homogeneous variety''', that is, a [[smooth scheme|smooth]] projective variety ''X'' over a field '''F''' with a [[transitive action]] of a [[reductive group]] ''G'' (and smooth stabilizer subgroup; that is no restriction for '''F''' of [[characteristic (algebra)|characteristic]] zero). If ''X'' has an '''F'''-[[rational point]], then it is isomorphic to ''G''/''P'' for some [[Borel subgroup|parabolic subgroup]] ''P'' of ''G''. A projective homogeneous variety may also be realised as the orbit of a [[highest weight]] vector in a projectivized [[group representation|representation]] of ''G''. The complex projective homogeneous varieties are the [[compact space|compact]] flat model spaces for [[Cartan connection#Parabolic Cartan connections|Cartan geometries]] of parabolic type. They are homogeneous [[Riemannian manifold]]s under any [[maximal compact subgroup]] of ''G'', and they are precisely the [[coadjoint orbit]]s of [[compact Lie group]]s.

Flag manifolds can be [[symmetric space]]s. Over the complex numbers, the corresponding flag manifolds are the [[Hermitian symmetric space]]s. Over the real numbers, an ''R''-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric ''R''-spaces.

==Flags in a vector space==

{{main|flag (linear algebra)}}

A flag in a finite dimensional vector space ''V'' over a field '''F''' is an increasing sequence of [[Linear subspace|subspace]]s, where "increasing" means each is a proper subspace of the next (see [[filtration (abstract algebra)|filtration]]):
:<math>\{0\} = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V.</math>
If we write the dim ''V''<sub>''i''</sub> = ''d''<sub>''i''</sub> then we have
:<math>0 = d_0 < d_1 < d_2 < \cdots < d_k = n,</math>
where ''n'' is the [[dimension (linear algebra)|dimension]] of ''V''. Hence, we must have ''k'' ≤ ''n''. A flag is called a ''complete flag'' if ''d''<sub>''i''</sub> = ''i'' for all ''i'', otherwise it is called a ''partial flag''. The ''signature'' of the flag is the sequence (''d''<sub>1</sub>, ..., ''d''<sub>''k''</sub>).

A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.

==Prototype: the complete flag variety==

According to basic results of [[linear algebra]], any two complete flags in an ''n''-dimensional vector space ''V'' over a field '''F''' are no different from each other from a geometric point of view. That is to say, the [[general linear group]] [[Group action (mathematics)|acts]] transitively on the set of all complete flags.

Fix an ordered [[basis (linear algebra)|basis]] for ''V'', identifying it with '''F'''<sup>''n''</sup>, whose general linear group is the group GL(''n'','''F''') of ''n'' × ''n'' invertible matrices. The standard flag associated with this basis is the one where the ''i''th subspace is spanned by the first ''i'' vectors of the basis. Relative to this basis, the [[stabilizer (group theory)|stabilizer]] of the standard flag is the [[group (mathematics)|group]] of nonsingular [[lower triangular matrix|lower triangular matrices]], which we denote by ''B''<sub>''n''</sub>. The complete flag variety can therefore be written as a [[homogeneous space]] GL(''n'','''F''') / ''B''<sub>''n''</sub>, which shows in particular that it has dimension ''n''(''n''&minus;1)/2 over '''F'''.

Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the [[special linear group]] SL(''n'','''F''') of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a [[Borel subgroup]].

If the field '''F''' is the real or complex numbers we can introduce an [[inner product]] on ''V'' such that the chosen basis is [[orthonormal]]. Any complete flag then splits into a direct sum of one-dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the [[homogeneous space]]
:<math>U(n)/T^n</math>
where U(''n'') is the [[unitary group]] and T<sup>''n''</sup> is the ''n''-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(''n'') replaced by the orthogonal group O(''n''), and T<sup>''n''</sup>  by the diagonal orthogonal matrices (which have diagonal entries ±1).

==Partial flag varieties==

The partial flag variety
:<math> F(d_1,d_2,\ldots d_k, \mathbb F)</math>
is the space of all flags of signature (''d''<sub>1</sub>, ''d''<sub>2</sub>, ... ''d''<sub>''k''</sub>) in a vector space ''V'' of dimension ''n'' = ''d''<sub>''k''</sub> over '''F'''. The complete flag variety is the special case that ''d''<sub>''i''</sub> = ''i'' for all ''i''. When ''k''=2, this is a [[Grassmannian]] of ''d''<sub>1</sub>-dimensional subspaces of ''V''.

This is a homogeneous space for the general linear group ''G'' of ''V'' over '''F'''. To be explicit, take ''V'' = '''F'''<sup>''n''</sup> so that ''G'' = GL(''n'','''F'''). The stabilizer of a flag of nested subspaces ''V''<sub>''i''</sub> of dimension ''d''<sub>''i''</sub> can be taken to be the group of nonsingular [[block matrix|block]] lower triangular matrices, where the dimensions of the blocks are ''n''<sub>''i''</sub> := ''d''<sub>''i''</sub> &minus; ''d''<sub>''i''&minus;1</sub> (with ''d''<sub>0</sub> = 0).

Restricting to matrices of determinant one, this is a parabolic subgroup ''P'' of SL(''n'','''F'''), and thus the partial flag variety is isomorphic to the homogeneous space SL(''n'','''F''')/''P''.

If '''F''' is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space
:<math> U(n)/U(n_1)\times\cdots \times U(n_k)</math>
in the complex case, or
:<math> O(n)/O(n_1)\times\cdots\times O(n_k)</math>
in the real case.

== Generalization to semisimple groups ==
The upper triangular matrices of determinant one are a Borel subgroup of SL(''n'','''F'''), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it.

Hence, more generally, if ''G'' is a [[semisimple group|semisimple]] [[linear algebraic group|algebraic]] or [[Lie group]], then the (generalized) flag variety for ''G'' is ''G''/''P'' where ''P'' is a parabolic subgroup of ''G''. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other.

The extension of the terminology "flag variety" is reasonable, because points of ''G''/''P'' can still be described using flags. When ''G'' is a [[classical Lie group|classical group]], such as a [[symplectic group]] or [[orthogonal group]], this is particularly transparent. If (''V'', ''&omega;'') is a [[symplectic vector space]] then a partial flag in ''V'' is ''[[isotropic]]'' if the symplectic form vanishes on proper subspaces of ''V'' in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(''V'',''&omega;''). For orthogonal groups there is a similar picture, with a couple of complications. First, if '''F''' is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the [[split orthogonal group]]s. Second, for vector spaces of even dimension 2''m'', isotropic subspaces of dimension ''m'' come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space.

==Cohomology==

If ''G'' is a compact, connected Lie group, it contains a [[maximal torus]] ''T'' and the space ''G''/''T'' of left cosets with the [[quotient topology]] is a compact real manifold. If ''H'' is any other closed, connected subgroup of ''G'' containing ''T'', then ''G''/''H'' is another compact real manifold. (Both are actually complex homogeneous spaces in a canonical way through [[Complexification (Lie group)#Complex structures on homogeneous spaces | complexification]].) 

The presence of a complex structure and [[Cellular homology|cellular (co)homology]] make it easy to see that the [[cohomology ring]] of ''G''/''H'' is concentrated in even degrees, but in fact, something much stronger can be said. Because ''G'' → ''G/H'' is a [[principal bundle | principal ''H''-bundle]], there exists a classifying map ''G''/''H'' → ''BH'' with target the [[classifying space]] ''BH''. If we replace ''G''/''H'' with the [[Equivariant cohomology#Homotopy quotient|homotopy quotient]] ''G''<sub>''H''</sub> in the sequence ''G'' → ''G/H'' → ''BH'', we obtain a principal ''G''-bundle called the [[Equivariant cohomology#Homotopy quotient|Borel fibration]] of the right multiplication action of ''H'' on ''G'', and we can use the cohomological [[Serre spectral sequence]] of this bundle to understand the fiber-restriction [[homomorphism]] ''H''*(''G''/''H'') → ''H''*(''G'')
and the characteristic map ''H''*(''BH'') → ''H''*(''G''/''H''), so called because its image, the ''characteristic subring'' of ''H''*(''G''/''H''), carries the [[characteristic class]]es of the original bundle ''H'' → ''G'' → ''G''/''H''.

Let us now restrict our coefficient ring to be a field ''k'' of characteristic zero, so that,
by [[Hopf algebra#Cohomology of Lie groups|Hopf's theorem]], ''H''*(''G'') is an [[exterior algebra]] on generators of odd degree (the subspace of [[Primitive element (co-algebra)|primitive elements]]). It follows that the edge homomorphisms 

:<math>E_{r+1}^{0,r} \to E_{r+1}^{r+1,0}</math>

of the spectral sequence must eventually take the space of primitive elements in the left column ''H''*(''G'') of the page ''E''<sub>2</sub> bijectively into the bottom row ''H''*(''BH''): we know ''G'' and ''H'' have the same [[Cartan subgroup | rank]],
so if the collection of edge homomorphisms were ''not'' full rank on the primitive subspace, then the image of the bottom row ''H''*(''BH'') in the final page ''H''*(''G''/''H'') of the sequence would be infinite-dimensional as a ''k''-vector space, which is impossible, for instance by [[cellular homology|cellular cohomology]] again, because a compact homogeneous space admits a finite [[CW complex|CW structure]].

Thus the ring map ''H''*(''G''/''H'') → ''H''*(''G'') is trivial in this case, and the characteristic map is surjective, so that ''H''*(''G''/''H'') is a quotient of ''H''*(''BH''). The kernel of the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements in the image of the canonical map ''H''*(''BG'') → ''H''*(''BH'') induced by the inclusion of ''H'' in ''G''. 

The map ''H''*(''BG'') → ''H''*(''BT'') is injective, and likewise for ''H'', with image the subring ''H''*(''BT'')<sup>''W''(''G'')</sup> of elements invariant under the action of the [[Weyl group]], so one finally obtains the concise description

:<math>H^*(G/H) \cong H^*(BT)^{W(H)}/\big(\widetilde{H}^*(BT)^{W(G)}\big),</math>

where <math>\widetilde H^*</math> denotes positive-degree elements and the parentheses the generation of an ideal. For example, for the complete complex flag manifold ''U''(''n'')/''T''<sup>''n''</sup>, one has 

:<math>H^*\big(U(n)/T^n\big) \cong \mathbb{Q}[t_1,\ldots,t_n]/(\sigma_1,\ldots,\sigma_n),</math>

where the ''t''<sub>''j''</sub> are of degree 2 and the σ<sub>''j''</sub> are the first ''n'' [[elementary symmetric polynomials]] in the variables ''t''<sub>''j''</sub>. For a more concrete example, take ''n'' = 2, so that ''U''(''2'')/[''U''(1) × ''U''(1)] is the complex [[Grassmannian]] Gr(1,<math>\mathbb{C}</math><sup>2</sup>) ≈ <math>\mathbb{C}</math>''P''<sup>1</sup> ≈ ''S''<sup>2</sup>. Then we expect the cohomology ring to be an exterior algebra on a generator of degree two (the [[fundamental class]]), and indeed,

:<math>H^*\big(U(2)/T^2\big) \cong \mathbb{Q}[t_1,t_2]/(t_1 + t_2, t_1 t_2)
 \cong \mathbb{Q}[t_1]/(t_1^2),</math>

as hoped.

==Highest weight orbits and projective homogeneous varieties==

If ''G'' is a semisimple algebraic group (or Lie group) and ''V'' is a (finite dimensional) highest weight representation of ''G'', then the highest weight space is a point in the [[projective space]] P(''V'') and its orbit under the action of ''G'' is a [[projective algebraic variety]]. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for ''G'' arises in this way.

[[Armand Borel]] showed{{Citation needed|reason=Need to cite the paper where Borel proves what follows|date=March 2021}} that this characterizes the flag varieties of a general semisimple algebraic group ''G'': they are precisely the [[complete variety|complete]] homogeneous spaces of ''G'', or equivalently (in this context), the projective homogeneous ''G''-varieties.

==Symmetric spaces==
{{main|Symmetric space}}
Let ''G'' be a semisimple Lie group with maximal compact subgroup ''K''. Then ''K'' acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety ''G''/''P'' is a compact homogeneous [[Riemannian manifold]] ''K''/(''K''&cap;''P'') with isometry group ''K''. Furthermore, if ''G'' is a complex Lie group, ''G''/''P'' is a homogeneous [[Kähler manifold]].

Turning this around, the Riemannian homogeneous spaces

:''M'' = ''K''/(''K''&cap;''P'')

admit a strictly larger Lie group of transformations, namely ''G''. Specializing to the case that ''M'' is a [[symmetric space]], this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano.

If ''G'' is a complex Lie group, the symmetric spaces ''M'' arising in this way are the compact [[Hermitian symmetric space]]s: ''K'' is the isometry group, and ''G'' is the biholomorphism group of ''M''.

Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are Riemannian symmetric spaces under ''K'' are known as symmetric R-spaces. The symmetric R-spaces which are not Hermitian symmetric are obtained by taking ''G'' to be a [[real form]] of the biholomorphism group ''G''<sup>c</sup> of a Hermitian symmetric space ''G''<sup>c</sup>/''P''<sup>c</sup> such that ''P'' := ''P''<sup>c</sup>&cap;''G'' is a parabolic subgroup of ''G''.  Examples include [[projective space]]s (with ''G'' the group of [[projective transformation]]s) and [[sphere]]s (with ''G'' the group of [[conformal transformation]]s).

==See also==

* [[Parabolic Lie algebra]]
* [[Bruhat decomposition]]

==References==

* Robert J. Baston and Michael G. Eastwood, ''The Penrose Transform: its Interaction with Representation Theory'', Oxford University Press, 1989.
* Jürgen Berndt, ''[http://www.mth.kcl.ac.uk/~berndt/sophia.pdf Lie group actions on manifolds]'', Lecture notes, Tokyo, 2002.
* Jürgen Berndt, Sergio Console and Carlos Olmos, ''[https://books.google.com/books?id=u3w4f63rmU8C Submanifolds and Holonomy]'', Chapman & Hall/CRC Press, 2003.
* Michel Brion, ''[http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html Lectures on the geometry of flag varieties]'', Lecture notes, Varsovie, 2003.
* [[James E. Humphreys]], ''[https://books.google.com/books?id=hNgRLxlwL8oC Linear Algebraic Groups]'', Graduate Texts in Mathematics, 21, Springer-Verlag, 1972.
* S. Kobayashi and T. Nagano, ''On filtered Lie algebras and geometric structures'' I, II, J. Math. Mech. '''13''' (1964), 875–907, '''14''' (1965) 513–521.

{{Authority control}}
[[Category:Differential geometry]]
[[Category:Algebraic homogeneous spaces]]