Flag Manifolds

Introduction

A flag in a vector space is a chain of subspaces, each contained in the next; a complete flag is a chain whose dimensions increase by one at each step, and the flag manifold is the set of all flags of a given type. It is the most general of the classical homogeneous spaces of the general linear group: the projective space is the flag manifold of lines, the Grassmannian is the flag manifold of a single subspace of a fixed dimension, and the general flag manifold collects a whole chain of nested subspaces at once. The group $GL(n, \mathbb{K})$ acts transitively on the flags, the stabiliser of a flag is a parabolic subgroup, and the quotient is a smooth projective variety — the flag variety — whose geometry organises the representation theory, the intersection theory and the cohomology of the linear group.

The flag manifold is the natural home of two classical theories. In geometry it is the universal family of nested subspaces, the base of the tautological flag of bundles, and the classifying space for the filtrations of a vector bundle; in representation theory it is the space whose line bundles and their sections realise the irreducible representations of the linear group, by the Borel–Weil theorem, and whose cohomology is the ring generated by the Chern classes of the tautological quotients. The combinatorial structure — the Bruhat decomposition into cells indexed by the symmetric group, the partial order on the permutations and the Schubert varieties — is the source of the Schubert calculus, and it is the model for the general theory of the flag manifolds $G/P$ of a semisimple group and a parabolic subgroup.

This article develops the flag manifolds of a vector space over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. It defines the complete and the partial flags, identifies the flag manifold with a quotient of the general linear group and of its maximal compact subgroup, and computes the dimension by the two methods; constructs the tautological flag of bundles and the quotient line bundles; gives the Bruhat decomposition into Schubert cells, proves that the cells are affine spaces indexed by the symmetric group and computes the Euler characteristic and the Borel presentation of the cohomology; describes the Schubert varieties, the Bruhat order and the incidence relations; gives the Plücker-type embedding of the flag manifold into a product of Grassmannians, so that it is a projective variety; and states the generalisation to the flag manifolds $G/P$ of a semisimple group, with the roles of the Weyl group, the roots and the parabolic subalgebras. The compact flag manifolds that are symmetric — the Grassmannians and the projective spaces — are read off against Symmetric Spaces.

The article assumes Smooth Manifolds and Differential Geometry for manifolds and quotients; Homogeneous Spaces for the quotient manifold theorem, the invariant metric and the isotropy representation; Grassmannians and Stiefel Manifolds for the Grassmannian, the tautological bundle and the Schubert cells there; Symmetric Spaces for the symmetric case; Fibre Bundles, Connections and Curvature for the tautological flag of bundles; The Exterior Algebra and The Determinant and Alternating Forms for the Plücker coordinates; from Part I and Part II, Groups, Group Actions and Structure, Matrix Groups and Classical Groups, Lie Groups and The Lie Correspondence and the Adjoint Representation for the Weyl group, the parabolic subgroups and the root data, all cited and not re-derived. The Kähler structure of the complex flag manifold is the subject of Kähler Geometry; the Schubert calculus belongs to the enumerative geometry of the algebraic side and is cited as standard; the homology and cohomology of the classical homogeneous spaces and the symplectic reading of the flag manifold as a coadjoint orbit lie outside this article. No physics is invoked.

Flags and the Flag Manifold

Definitions

Definition. Let $V$ be a finite-dimensional vector space over $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, of dimension $n$. A flag in $V$ is a strictly increasing chain of subspaces

$$ 0 = V_0 \subset V_1 \subset V_2 \subset \cdots \subset V_k \subset V_{k+1} = V , $$

of prescribed dimensions $\dim V_i = d_i$ with $0 < d_1 < \cdots < d_k < n$. The flag is complete if $k = n-1$ and $d_i = i$ for all $i$; otherwise it is partial of type $(d_1, \ldots, d_k)$.

Definition. For a type $(d_1, \ldots, d_k)$ with $1 \leq d_1 < \cdots < d_k \leq n-1$, the flag manifold is

$$ F\ell(d_1, \ldots, d_k; V) = \{(V_1, \ldots, V_k) : 0 \subset V_1 \subset \cdots \subset V_k \subset V,\ \dim V_i = d_i\}, $$

and the complete flag manifold is $F\ell_n(\mathbb{K}) = F\ell(1, 2, \ldots, n-1; \mathbb{K}^n)$. When $k = 1$ the flag manifold is the Grassmannian $\mathrm{Gr}_{d_1}(V)$.

Example (the smallest cases). $F\ell_2(\mathbb{K}) = \mathbb{KP}^1$, the manifold of lines in $\mathbb{K}^2$, and $F\ell_1(\mathbb{K})$ is a point; $F\ell_3(\mathbb{K}) = F\ell(1,2;\mathbb{K}^3)$ is the manifold of pairs (line, plane) with the line contained in the plane, which is the incidence variety of points and lines of $\mathbb{KP}^2$, of real dimension $3\dim_{\mathbb{R}}\mathbb{K}$; $F\ell(1; \mathbb{K}^n) = \mathbb{KP}^{n-1}$; $F\ell(k; \mathbb{K}^n) = \mathrm{Gr}_k(\mathbb{K}^n)$.

The Homogeneous Space Structure

Theorem. The general linear group $GL(n, \mathbb{K})$ acts transitively on $F\ell(d_1, \ldots, d_k; \mathbb{K}^n)$ by acting on each subspace of the flag. The stabiliser of the standard flag

$$ 0 \subset \langle e_1 \rangle \subset \langle e_1, e_2\rangle \subset \cdots \subset \langle e_1, \ldots, e_{d_k}\rangle \subset \mathbb{K}^n $$

is the group $P$ of block upper triangular matrices, with blocks of sizes $d_1, d_2 - d_1, \ldots, d_k - d_{k-1}, n - d_k$; for the complete flag it is the group $B$ of invertible upper triangular matrices, the Borel subgroup. Hence

$$ F\ell(d_1, \ldots, d_k; \mathbb{K}^n) = GL(n, \mathbb{K})/P , \qquad F\ell_n(\mathbb{K}) = GL(n, \mathbb{K})/B , $$

and, replacing the general linear group by its maximal compact subgroup, $GL(n, \mathbb{K})$ by $U(n)$ over $\mathbb{C}$ and by $O(n)$ over $\mathbb{R}$,

$$ F\ell_n(\mathbb{C}) = U(n)/T $$

with $T$ the diagonal maximal torus.

Proof sketch. Transitivity: given two flags there is a basis adapted to each, and the change-of-basis matrix carries one to the other. The stabiliser of the standard flag is the group preserving each subspace of the chain, which is exactly the block upper triangular group. The compact form follows because $U(n)$ acts transitively on the flags of $\mathbb{C}^n$ and the stabiliser of the standard flag in $U(n)$ is the diagonal torus $T \cong U(1)^n$.

Theorem (dimension). The flag manifold has dimension

$$ \dim_{\mathbb{R}} F\ell(d_1, \ldots, d_k;\mathbb{K}^n) = (\dim_{\mathbb{R}}\mathbb{K})\sum_{i=0}^{k} d_i\,(d_{i+1} - d_i), \qquad d_0 = 0, \quad d_{k+1} = n , $$

as a real manifold; in particular

$$ \dim_{\mathbb{R}} F\ell_n(\mathbb{R}) = \frac{n(n-1)}{2}, \qquad \dim_{\mathbb{R}} F\ell_n(\mathbb{C}) = n(n-1), \qquad \dim_{\mathbb{C}} F\ell_n(\mathbb{C}) = \frac{n(n-1)}{2}. $$

Proof sketch. This is the dimension count $\dim GL(n,\mathbb{K}) - \dim P$, and the sum formula is the count of the entries of the block upper triangular matrix that are not forced to vanish: the $i$-th off-diagonal block of size $d_i \times (d_{i+1} - d_i)$. For the complete flag over $\mathbb{C}$, $\dim_\mathbb{R} = n^2 - n = n(n-1)$, of complex dimension $n(n-1)/2$. For the Grassmannian, $k=1$ and the sum is $d_1(n - d_1)$.

Corollary. The flag manifold is a compact, connected smooth manifold; it is a projective variety in the complex case; and it is simply connected over $\mathbb{C}$, being a homogeneous space of the simply connected group $SU(n)$ (the quotient of $U(n)$ by the connected torus is the same manifold).

The Tautological Bundles

The Tautological Flag

Definition. Over the complete flag manifold $F\ell_n(\mathbb{K})$ let $\mathcal{S}_i$ be the subbundle of the trivial bundle $\mathbb{K}^n$ whose fibre at the flag $V_\bullet$ is the subspace $V_i$; the chain

$$ 0 \subset \mathcal{S}_1 \subset \mathcal{S}_2 \subset \cdots \subset \mathcal{S}_{n-1} \subset \mathbb{K}^n $$

is the tautological flag, and the line bundles $\mathcal{L}_i = \mathcal{S}_i/\mathcal{S}_{i-1}$ for $i = 1, \ldots, n$ (with $\mathcal{S}_0 = 0$, $\mathcal{S}_n = \mathbb{K}^n$) are the tautological quotient lines. The flag manifold is the classifying space of complete flags of a rank-$n$ vector bundle, and the tautological flag is the universal family.

Theorem. The bundles $\mathcal{S}_i$ and $\mathcal{L}_i$ are smooth vector bundles; the bundle $\mathcal{S}_i$ has rank $i$; the line bundles satisfy $\mathcal{L}_1 \oplus \cdots \oplus \mathcal{L}_n \cong \mathbb{K}^n$ (the trivial bundle) in the sense of the associated graded; and the tangent bundle of the flag manifold is

$$ T F\ell_n(\mathbb{K}) \cong \bigoplus_{1 \leq i < j \leq n} \operatorname{Hom}(\mathcal{L}_i, \mathcal{L}_j) = \bigoplus_{i

of rank $\sum_{i

Proof sketch. The fibre of $\mathcal{S}_i$ varies smoothly by the local triviality of the flag manifold; the tangent space at a flag is the space of infinitesimal deformations of the chain, which is the space of filtered endomorphisms, identified with the direct sum of the $\operatorname{Hom}(\mathcal{L}_i,\mathcal{L}_j)$; the rank count gives the dimension of the previous theorem. The bundle theory is that of Fibre Bundles, Connections and Curvature.

Corollary (the characterisation of flags). A rank-$n$ bundle $E$ over a paracompact base together with a complete flag of subbundles is classified by a map into $F\ell_n(\mathbb{K})$; the tautological flag pulls back to the given flag. In particular the projective space classifies the line subbundles of a rank-$n$ bundle, and the flag manifold classifies the filtrations.

Example. For $n = 2$ the flag manifold is $F\ell_2 = \mathbb{KP}^1 = S^{\dim_{\mathbb{R}}\mathbb{K}}$, and the tautological flag is $0 \subset \mathcal{L}_1 \subset \mathbb{K}^2$ with $\mathcal{L}_1$ the tautological line bundle; the tangent bundle is $\mathcal{L}_1^{\vee}\otimes\mathcal{L}_2 \cong \mathcal{L}_1^{-2}$, whose degree is $-2$ and whose total space is the real tangent bundle of the sphere after the real structure is taken. This is the base case of the general statement.

The Bruhat Decomposition

Schubert Cells

Definition. Fix the standard flag and let $B$ be the Borel subgroup of upper triangular matrices. The Bruhat decomposition of the complete flag manifold is the orbit decomposition under $B$,

$$ F\ell_n(\mathbb{K}) = \bigsqcup_{w \in S_n} B \cdot \sigma_w , $$

where $S_n$ is the symmetric group (the Weyl group of $GL(n)$) and $\sigma_w$ is the flag whose $i$-th subspace is spanned by the coordinate vectors permuted by $w$. The orbit $C_w = B\cdot\sigma_w$ is the Schubert cell of $w$.

Theorem (Bruhat). The Schubert cell $C_w$ is isomorphic to an affine space of dimension $\ell(w)$, the number of inversions of the permutation $w$; consequently

$$ F\ell_n(\mathbb{C}) = \bigsqcup_{w \in S_n} \mathbb{C}^{\ell(w)}, \qquad \chi(F\ell_n(\mathbb{C})) = \sum_{w\in S_n}(-1)^{2\ell(w)} = n! , $$

the cells all having even real dimension $2\ell(w)$.

Proof sketch. In terms of the matrix of a flag in the standard basis, the cell $C_w$ is described by the vanishing of certain entries and the value $1$ at the pivots prescribed by $w$; the free entries are exactly the inversions of $w$, so the cell is an affine space of dimension $\ell(w)$. The computation of the Euler characteristic is the alternating sum over the cells, and $2\ell(w)$ is even, so every term contributes $+1$ and the sum is the order of $S_n$.

Example (the Schubert cells of $F\ell_3(\mathbb{C})$). There are $3! = 6$ cells, one for each permutation, of complex dimensions $0, 1, 1, 2, 2, 3$ corresponding to the permutations $123, 132, 213, 231, 312, 321$ (the longer permutations having the larger dimension), and $\chi = 6$. The partial flag manifold $F\ell(1;\mathbb{C}^3) = \mathbb{CP}^2$ has the three cells of dimensions $0,1,2$ corresponding to the three cosets $S_3/S_2$, matching the three projective Schubert cells of Grassmannians and Stiefel Manifolds.

Definition. The closure $X_w = \overline{C_w}$ is the Schubert variety of $w$, and the Bruhat order on $S_n$ is defined by $v \leq w$ if $X_v \subseteq X_w$. The length $\ell(w)$ is the number of inversions, and the Chevalley formula computes the product of the class of a divisor with a Schubert class in the cohomology ring.

Proposition. The Schubert variety $X_w$ is the set of flags whose $i$-th subspace has intersection of dimension at least the number prescribed by $w$ with the standard $j$-th coordinate subspace; it is a closed subvariety of dimension $\ell(w)$, and $X_v \subseteq X_w$ if and only if $v \leq w$ in the Bruhat order.

Proof sketch. The intersection conditions are closed conditions; the cell of $v$ lies in the variety of $w$ exactly when the vanishing pattern of $v$ is weaker than that of $w$, which is the rank condition of the Bruhat order.

The Cohomology Ring

Theorem (Borel presentation). The integral cohomology ring of the complete flag manifold is

$$ H^*(F\ell_n(\mathbb{C});\mathbb{Z}) \cong \mathbb{Z}[x_1, \ldots, x_n]\big/\bigl(e_1(x), \ldots, e_n(x)\bigr), $$

where $e_i$ is the $i$-th elementary symmetric polynomial and $x_i$ is the first Chern class of the tautological quotient line $\mathcal{L}_i$; the ring is free of rank $n!$ as a $\mathbb{Z}$-module, with basis the Schubert classes $[X_w]$ for $w \in S_n$.

Proof sketch. The quotient $U(n)/T$ has classifying map that identifies $H^*(F\ell_n)$ with the $T$-equivariant cohomology of a point modulo the ideal generated by the elementary symmetric classes, which is the Borel presentation of the cohomology of a homogeneous space of a compact group; the Schubert cells give a basis by the cell decomposition.

Corollary (Euler characteristic and Poincaré polynomial). The cohomology is concentrated in even degrees, the Poincaré polynomial is the statistic of the inversion number,

$$ P_t(F\ell_n(\mathbb{C})) = \sum_{w\in S_n} t^{2\ell(w)} = \prod_{i=1}^{n}\frac{t^{2i}-1}{t^2-1} = [n]_{t^2}! , $$

the $q$-factorial, and the Euler characteristic is $[n]_1! = n!$.

Example. For $n = 3$ the Poincaré polynomial is $1 + 2t^2 + 2t^4 + t^6$, and the cohomology is generated by $x_1, x_2, x_3$ modulo $e_1, e_2, e_3$. For $n = 4$ the polynomial is $1 + 3t^2 + 5t^4 + 6t^6 + 5t^8 + 3t^{10} + t^{12}$, whose coefficients are the numbers of permutations of $S_4$ with $0, 1, \ldots, 6$ inversions.

Remark (Schubert calculus). The product in the cohomology ring, expressed in the basis of the Schubert classes, has structure constants given by the Littlewood–Richardson coefficients; this is the Schubert calculus of the flag manifold, and it is the general form of the calculus on the Grassmannian. It belongs to the enumerative and algebraic geometry of the corpus and is quoted here as standard.

The Plücker Embedding

Theorem. The flag manifold $F\ell(d_1, \ldots, d_k; V)$ embeds in the product of the Grassmannians,

$$ F\ell(d_1, \ldots, d_k; V) \longrightarrow \mathrm{Gr}_{d_1}(V)\times\mathrm{Gr}_{d_2}(V)\times\cdots\times\mathrm{Gr}_{d_k}(V), $$

by the map sending a flag to its sequence of subspaces; the image is the closed subvariety defined by the incidence relations $V_{i} \subset V_{i+1}$, and composing with the Plücker embeddings of the factors gives an embedding of the flag manifold into a product of projective spaces, so the flag manifold over $\mathbb{C}$ is a smooth projective variety. The Plücker coordinates of a flag are the Plücker coordinates of its subspaces, subject to the incidence conditions.

Proof sketch. The map is injective because a flag is its sequence of subspaces; it is an immersion because each subspace varies smoothly; and the image is closed because the incidence conditions are the vanishing of the minors of the matrices expressing one subspace in the basis of the next, which are polynomial equations. The Plücker description of each factor is that of Grassmannians and Stiefel Manifolds.

Corollary (line bundles and the Borel–Weil theorem). Every holomorphic line bundle on $F\ell_n(\mathbb{C})$ is a product of powers of the tautological quotient lines, $\mathcal{O}(a_1, \ldots, a_n) = \mathcal{L}_1^{a_1}\otimes\cdots\otimes\mathcal{L}_n^{a_n}$, the expression being defined up to the relation $\mathcal{L}_1\otimes\cdots\otimes\mathcal{L}_n \cong \mathcal{O}$; the finite-dimensional irreducible holomorphic representations of $GL(n,\mathbb{C})$ are realised on the spaces of sections $H^0(F\ell_n(\mathbb{C}), \mathcal{O}(a_1,\ldots,a_n))$ for dominant weights $(a_1 \geq \cdots \geq a_n)$, by the Borel–Weil theorem.

Proof sketch. The Picard group is computed from the homogeneous space description: the characters of the torus $T$ index the line bundles, and the dominant characters are those with nonincreasing exponents; the sections are the polynomials of the prescribed flag-multi-degree, giving the irreducible representation of highest weight.

Remark. This is the geometric form of the classification of the representations of the general linear group by highest weights, and the flag manifold is the universal space on which the representations of all dominant weights are simultaneously visible as spaces of sections of line bundles. The representation theory itself is the corpus's algebraic material and is cited as standard; the cohomology of the flag manifold, with its Schubert basis, is the topological shadow.

Generalised Flag Manifolds

Definition. Let $G$ be a semisimple Lie group with Lie algebra $\mathrm{G}$, and let $P$ be a parabolic subgroup, that is, a closed subgroup containing a Borel subgroup; equivalently, $P$ is the normaliser of a parabolic subalgebra $\mathrm{P}$, which is a subalgebra containing a Borel subalgebra $\mathrm{B}$. The quotient

$$ G/P $$

is the generalised flag manifold of the pair $(G, P)$; when $G = GL(n,\mathbb{C})$ and $P$ is the stabiliser of a chain of subspaces, this is the flag manifold of the previous sections.

Theorem. A generalised flag manifold $G/P$ is a compact complex manifold, in fact a smooth projective variety, homogeneous under the action of $G$; its tangent space at the identity coset is $\mathrm{G}/\mathrm{P}$, and its cohomology has a basis of Schubert classes indexed by the Weyl group $W$ of $G$ modulo the parabolic subgroup $W_P$, with a Borel presentation

$$ H^*(G/P;\mathbb{Z}) \cong \mathbb{Z}[\mathrm{T}^*]\big/\bigl(\text{the Weyl-invariant polynomials with positive degree}\bigr) $$

for the maximal compact quotient, where $\mathrm{T}$ is the Cartan subalgebra.

Proof sketch. The compact form of $G$ acts transitively with stabiliser the compact form of $P$, giving a compact complex homogeneous space; the embedding in a projective space is given by a dominant weight whose stabiliser is $P$, by the Borel–Weil theorem; the cohomology statement is the Borel presentation for a homogeneous space of a compact group, and the Schubert classes come from the Bruhat decomposition of $G$ into the double cosets $BwB$.

Example (the incidence variety and the projective space). The projective space $\mathbb{KP}^{n-1} = GL(n)/P_1$, with $P_1$ the stabiliser of a line, is the flag manifold of a maximal parabolic; the Grassmannian $\mathrm{Gr}_k(\mathbb{K}^n) = GL(n)/P_k$ is the flag manifold of another; and the complete flag $GL(n)/B$ is the flag manifold of the Borel. In the semisimple generalisation the role of the symmetric group is taken by the Weyl group $W$, and the role of the parabolic subgroup of $S_n$ by the parabolic subgroup $W_P$ of $W$ generated by the reflections in the simple roots not in $P$.

Theorem (the symmetric case). A compact generalised flag manifold $G/P$ that is a Riemannian symmetric space is necessarily a Hermitian symmetric space, because it is a complex manifold and the complex structure is invariant under the action of the compact group; conversely every compact Hermitian symmetric space is a generalised flag manifold $G/P$ with $P$ a maximal parabolic of Hermitian type. The symmetric flag manifolds are therefore exactly the compact Hermitian symmetric spaces: the complex Grassmannians, the projective spaces over $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$, the quadrics, the Lagrangian Grassmannians and the two exceptional cases $E_6/(\mathrm{Spin}(10)\cdot U(1))$ and $E_7/(E_6\cdot U(1))$. A flag manifold that is not of this type is homogeneous without being symmetric, and its canonical connection has nonzero torsion, the component of the bracket in $\mathrm{M}$ of Homogeneous Spaces; the complete flag manifold $U(n)/T$ with $n \geq 3$ is the basic example, its isotropy representation being the sum of the positive root spaces, whose brackets are not contained in the torus.

Proof sketch. A complex structure on a compact symmetric space $G/H$ is invariant under $G$ and hence parallel, so the symmetric space is Hermitian; the isotropy algebra of a Hermitian symmetric space has a central circle, which identifies the symmetric subgroup $H$ as a maximal parabolic with the right root-theoretic property, and the classification of the compact Hermitian symmetric spaces then gives the list. For $U(n)/T$ with $n \ge 3$ the roots $\alpha, \beta$ with $\alpha+\beta$ a root give a bracket $[\mathrm{M},\mathrm{M}]$ with a nonzero component in $\mathrm{M}$, so the symmetric-pair condition fails.

Remark (the coadjoint orbit picture). The generalised flag manifolds are the coadjoint orbits of the compact form of $G$ through the regular elements, and they carry the natural symplectic structure of the orbit; the symplectic geometry of the orbits, the moment maps and the quantisation are Symplectic Geometry, written in parallel, and only the homogeneous and combinatorial structure is used here.

Summary

A flag in a vector space is a strictly increasing chain of subspaces, and the flag manifold is the set of all flags of a fixed type. The general linear group acts transitively, the stabiliser of the standard flag is a parabolic subgroup — the Borel subgroup in the complete case — so the flag manifold is the quotient $GL(n,\mathbb{K})/P$, with compact form $F\ell_n(\mathbb{C}) = U(n)/T$ and dimension $(\dim_\mathbb{R}\mathbb{K})\sum_i d_i(d_{i+1}-d_i)$; the complete flag manifold over $\mathbb{C}$ has complex dimension $n(n-1)/2$, and in dimension two it is the sphere of dimension $\dim_\mathbb{R}\mathbb{K}$.

Over the flag manifold runs the tautological flag $\mathcal{S}_1 \subset \cdots \subset \mathcal{S}_{n-1}$, with quotient lines $\mathcal{L}_i$; the flag manifold classifies the complete flags of subbundles of a rank-$n$ bundle, and the tangent bundle is the direct sum of the $\operatorname{Hom}(\mathcal{L}_i, \mathcal{L}_j)$ over $i < j$. The Bruhat decomposition divides the complete flag manifold into Schubert cells indexed by the symmetric group, the cell of a permutation being an affine space of dimension equal to the number of its inversions; all cells have even real dimension, so the Euler characteristic is $n!$, and the cohomology has the Borel presentation $\mathbb{Z}[x_1,\ldots,x_n]/(e_1,\ldots,e_n)$ with the Schubert classes as a basis. The Schubert varieties are the closures of the cells, ordered by the Bruhat order, and their intersection theory is the Schubert calculus.

The flag manifold embeds in a product of Grassmannians by the sequence of subspaces of a flag, with image cut out by the incidence relations; composed with the Plücker embeddings this exhibits it as a smooth projective variety, and every line bundle on it is a product of powers of the tautological quotient lines, whose sections realise the irreducible representations by the Borel–Weil theorem. The generalised flag manifolds $G/P$ of a semisimple group and a parabolic subgroup carry the same structure with the symmetric group replaced by the Weyl group and the Schubert cells by the double cosets $BwB$; the compact generalised flag manifolds that are Riemannian symmetric are exactly the maximal parabolics with connected complement in the Dynkin diagram, that is the Grassmannians, the projective spaces and the quadrics, and the remaining flag manifolds are homogeneous without being symmetric. The Kähler geometry, the Schubert calculus and the cohomology of the classical homogeneous spaces are treated in Kähler Geometry.

Summary of Notation

Further Reading

• Michel Demazure, "Désingularisation des variétés de Schubert généralisées", Annales scientifiques de l'École Normale Supérieure 7 (1974), 53–88, for the Schubert varieties and the Demazure resolution.
• William Fulton, Young Tableaux (Cambridge University Press, 1997), for the Schubert cells, the Bruhat order and the Schubert calculus.
• Raoul Bott, "An application of the Morse theory to the topology of Lie groups", Bulletin de la Société Mathématique de France 84 (1956), 251–281, for the cell decomposition and the cohomology of the flag manifolds.
• Armand Borel, "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts", Annals of Mathematics 57 (1953), 115–207, for the Borel presentation of the cohomology.
• Ira M. Gelfand and Mikhail A. Naimark, Unitary Representations of the Classical Groups (Deutsche Verlag der Wissenschaften, 1957), for the representation-theoretic use of the flag manifolds.
• James E. Humphreys, Linear Algebraic Groups (Springer, 1975), for the parabolic subgroups, the Bruhat decomposition and the generalised flag manifolds.
• Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, 1978), for the homogeneous and symmetric structure of the generalised flag manifolds.