Questions tagged [flag-varieties]
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151 questions
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How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
- 9,009
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0
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268
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What's the relation of the Hecke algebra of a pair and the flag variety?
Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively.
Then the Hecke algebra ...
- 9,009
9
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2
answers
964
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Hard Lefschetz Theorem for the Flag Manifolds
In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...
- 163
1
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1
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Volume of a GIT quotient of projective lines
Say $X= \mathbb{P^1}\times \cdots \times \mathbb{P}^1$ is a product of $n\geq3$ lines. Let the group $G=\text{SL}(2)$ act on $X$ diagonally, and let $\mathcal{L} = \mathcal{L}(a_1,\ldots,a_n)$ be the ...
10
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1
answer
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Examples of Symplectic Questions Solved by ``Mirror Symmetry Translation'' to Complex Questions
According to the proponents of homological mirror symmetry, when a complex and symplectic manifold are mirror symmetric, we can take difficult questions about the symplectic space and transfer them ...
- 1,453
5
votes
1
answer
555
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Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and $S^-...
- 1,719
4
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2
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820
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Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles
The Borel--Bott--Weil Theorem gives the dimensions of the cohomology groups of the equivariant line bundles over flag manifolds. Does there exist an analogous result for general equivariant vector ...
- 357
7
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2
answers
1k
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What is the Explicit Relationship between Coadjoint Orbits and Flag Manifolds?
Given a complex semi-simple Lie group $G$, it acts smoothly on the dual $\frak{g}^*$ of its Lie algebra $\frak{g}$ by the coadjoint action. The orbits of that action are called coadjoint orbits.
A ...
- 155
3
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2
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476
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Specialisations of flag varieties
Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group.
My question concerns specialisations of flag ...
- 21.3k
5
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1
answer
312
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Non-vanishing of elements in cohomology of full Flag varieties
Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$:
$0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
- 14.3k
1
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1
answer
175
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When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?
Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and
$\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...
user21574
3
votes
2
answers
325
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Why is $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the space of complete flags $GL_3/B$?
In one the the answers to this thread " Can one embedd the projectivezed tangent space of CP^2 in a projective space? " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety ...
- 131
8
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1
answer
566
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Quotienting $SU(3)$ by $U(1)$?
As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
- 155
1
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0
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394
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Pullback of a sheaf associated to a divisor
I am reading a paper Desingularisation des varietes de Schubert generalisees by Demazure. I am interested in Lemma 3 on page 58. In particular, I would like to know whether the lemma is true and how ...
6
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3
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441
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Closed orbits of complete flags in $\mathbb{C}^n$
Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the Zariski-...
- 174
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4
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1k
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Automorphism group of flag manifolds?
If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms.
...
- 1,062
6
votes
1
answer
780
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Moduli of flag varieties
I work over an algebraically closed field $k$ of characteristic zero.
Recall that a flag variety is a projective variety which is a homogeneous space for some semisimple algebraic group. Every flag ...
- 21.3k
4
votes
1
answer
685
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Full exceptional set on flag variety
Define a full exceptional set in a triangulated category to be a partially ordered set of objects $\Delta_i$ which generate the category and such that $\text{Ext}^\bullet(\Delta_i, \Delta_j) = 0$ ...
- 1,423
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1
answer
201
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Non-equivariant vector bundles over complex projective $N$-space
From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist non-...
- 357
8
votes
1
answer
221
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Cominuscule property of nilpotent orbits
Let $G$ be a complex reductive Lie group, $G/P$ a flag manifold,
and $\Phi: T^* G/P \to {\mathfrak g}^*$ the moment map. So $\Phi(T^* G/P)$ is the closure of a nilpotent orbit.
Lots of classes of ...
- 27.8k
12
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2
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3k
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Cohomology ring of a flag variety and representation theory
I'm interested in the cohomology ring $H^*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to ...
1
vote
1
answer
200
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Variety of factorizations of differential operator
Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...
- 23
4
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1
answer
379
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Stalks of intersection cohomology complexes of Schubert varieties and Bruhat order
All varieties are over $\mathbb{C}$.
Let $G$ be a connected reductive group, $B\subseteq G$ a Borel subgroup.
Let $O_w$ be a $B$-orbit in $G/B$. I.e., $O_w$ is a Bruhat cell. In particular, it is ...
- 1,595
2
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0
answers
164
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Flags of varieties
I was wondering if there is a generalization of flags in the following way: Suppose you have a series of inclusions of affine varieties $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ ...
- 928
4
votes
1
answer
407
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How does the grading on the cohomology of a flag variety break up the regular representation of W?
The complex cohomology H^* of the manifold of flags in C^n is a quotient of C[x1,...,xn] by the ideal generated by symmetric polynomials with no constant term. In particular it has an action of the ...
- 824
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2
answers
2k
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Partial (or complete) flag varieties as GIT quotients of affine spaces
I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...
- 142
6
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466
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Purity and six operations?
The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.
For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity ...
- 13.2k
1
vote
1
answer
686
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Computing relative Lie algebra cohomology (as appears in Borel-Weil-Bott theorem)
Suppose $G$ is a complex Lie group, $P$ a Borel subgroup, $E$ a representation of $P$ that induces a vector bundle ${\cal E}$ over $G/P$. The general version of Borel-Weil-Bott theorem, as stated in ...
- 131
5
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1
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392
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Equivariant $K$-theory, singular vectors, and flag manifolds
For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
- 3,399
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175
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(semi-)Small resolutions of Peterson varieties
Peterson varieties (in type A) can be described as the subvarieties of the full flag variety
$$\{(F_{i})\;|\; F_{i}\subset \mathbb{C}^{n}, \; \dim F_{i} =i,\; N(F_{i})\subset F_{i+1}\}$$
where $N$ ...
- 1,201
2
votes
2
answers
613
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Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?
As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 \...
- 389
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2
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2k
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Defining Equations of a Flag Variety
I've been reading Fulton's Young Tableaux, and I'm trying to understand flag varieties. I want to understand the defining equations of a Flag Variety, but the coordinates in Fulton's Plucker relations,...
- 131
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0
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456
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Flag Varieties via Quiver Varieties
In type $A$ it is possible to realise the flag variety $\mathcal{F}$ of $\text{SL}_{n}(\mathbb{C})$ via Nakajima's quiver varieties: consider the vectors $v=(1,2,\ldots, n-1), w=(0,\ldots,0,n)$. Then, ...
- 1,201
4
votes
2
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372
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Kahler Metric Fundamental Forms and Cohomology Ring Generators
For the projective line $CP^1$, its cohomology ring has a single generator. Moreover, this generator is given by the cohomology class of the fundamental form associated associated to the Fubini--Study ...
- 263
2
votes
0
answers
385
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Weight decomposition and eigenspaces Euler vector field
Let $V$ be a finite dimensional vector space over $\mathbb{C}$, denote $V^{\times} = V -0$ and let $\pi : V^{\times} \rightarrow \mathbb{P}(V)$ be the quotient by the natural action of $\mathbb{C}^{\...
- 213
31
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3
answers
3k
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Rep Theory Consequences of Bott--Weil--Borel
I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
- 3,399
6
votes
0
answers
304
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Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)
For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...
- 3,399
6
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1
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674
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Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups
I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
- 3,399
3
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0
answers
163
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How to deduce (8.1) in Lusztig's "Equivariant K-theory and representations of Hecke Algebras"
Let G be a connected complex algebraic group G and X=G/B, where B is the Borel subgroup. $ M=G\times C^{*}$, where $C^*$ acts on X trivially. Let $K_M(X)$ be the equivariant K-theory. Let $s\in S$ be ...
- 31
2
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1
answer
256
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The real group orbits on the flag variety always contains the holomorphic directions?
Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\...
- 9,009
4
votes
0
answers
323
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The proof of the splitting principle in equivariant K-theory via flag manifolds
In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:
Let $j: T\...
- 9,009
20
votes
2
answers
4k
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What is the significance that the Springer resolution is a moment map?
Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution
$$
\mu: T^*\mathcal{B}\rightarrow \mathcal{N}
$$
is the moment ...
- 9,009
7
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0
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Is the Springer resolution a blow-up?
Let's consider the Springer resolution of the nilpotent cone $\mathcal{N}$ of a complex semisimple Lie algebra $\mathfrak{g}$, which is
$$
\widetilde{\mathcal{N}}=T^*\mathcal{B}\rightarrow \mathcal{N}...
- 9,009
5
votes
1
answer
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What kind of algebra has geometric realization as in "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups"
In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra $\...
- 9,009
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2
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2k
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The anticanonical bundle on a flag variety is ample
Hello,
I would like to get references or answers, for the following. How do I show that the anti-canonical line bundle (i.e. dual to top wedge power of cotangent bundle) on a flag variety (of a ...
- 5,562
4
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1
answer
347
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What is known about formality of flag varieties?
Let $G$ be a connected, complex reductive algebraic group and $X=G/P$ a (partial) flag variety. For example by "Real homotopy theory of Kähler manifolds", we know that its cohomology with real ...
- 13.2k
1
vote
1
answer
572
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moduli problem for flag varieties?
Hi,
Suppose $G$ is a reductive group over an algebraiclly closed field $k$
(suppose $k$ of char zero if you want at first). Let $X$ be its flag variety.
Question: What is the moduli problem that $X$ ...
- 2,842
13
votes
4
answers
2k
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Riemannian metric on a flag variety
$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = \...
- 7,273
7
votes
2
answers
1k
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Hecke algebra and $H^*(G/B)$
Given a complex reductive group, with Weyl group $W$, one can associate to it lots of "algebras of size $|W|$". For example $B$ equivariant functions on $G/B$ with convolution, grothendieck groups of ...
- 13.2k
65
votes
6
answers
9k
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Origin of terms "flag", "flag manifold", "flag variety"?
These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...
- 52.9k