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Questions tagged [flag-varieties]

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6
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1answer
135 views

Irreducibility of Gelfand-Serganova strata

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...
6
votes
1answer
240 views

Closures of torus orbits in flag varieties

Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$. Now, as far as I ...
4
votes
1answer
129 views

Equivalence of complex structures on flag manifold

Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. One way to construct a complex structure on $G/T$ is to choose a nilpotent subalgebra $\mathfrak{n}^+$ corresponding to some choice of ...
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0answers
32 views

Flag variety and Schubert cells for free modules

Let $R$ a principal domain, and $M$ a free module of rank $d$ on $R$, with a fixed basis $(m_1, \ldots, m_d)$. The definition of flags translates directly to this setting: a flag of $M$ would simply ...
6
votes
2answers
266 views

Degree of hypersurfaces and flag varieties

In the book Discriminants, Resultants, and Multidimensional Determinants of Andrei Zelevinsky,M.M. Kapranov and Izrail' Moiseevič Gel'fand, the authors give the following definition of degree of a ...
3
votes
1answer
140 views

Degeneration of coadjoint orbits

Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...
0
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0answers
150 views

Number of Generators of the Cohomology Ring of the Grassmannians

For complex projective space, its cohomology ring has $1$ generator. Extending up to the first Grassmannian which is not a projective space, that is, Grass$(4,2)$, a direct investigation shows that it ...
1
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0answers
112 views

Algebra Invariants of Schubert Calculus

For the Grassmannian Gr[N,k] of $k$-planes in $\mathbb{C}^N$, the cohomology ring $H^*(Gr[N,k])$ is a much studied object in an area called Schubert calculus. As a complex algebra, $H^*(Gr[N,k])$ is ...
14
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3answers
414 views

Schubert calculus expressed in terms of the cotangent space of the Grassmannians

Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,...
2
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0answers
27 views

Reference Request: Neighbor-avoiding sets of flags

Let $K$ be a finite field and $\mathcal{F}$ the set of full flags in $K^n$, i.e. $\mathcal{F}$ consists of chains of subspaces $$0 < V_1 < V_2 < V_3 < \ldots < V_{n-1} < K^n$$ such ...
5
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0answers
107 views

Fubini--Study Orthogonality for Schubert Calculus

Consider the following points: $\bullet$ Let ${\cal Harm}(n,d)$ denote the harmonic forms of the de Rham complex of the Grassmannian $Gr_{\mathbb{C}}(n,d)$ with respect to the Riemannian metric ...
5
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2answers
162 views

Positivity of coefficients of a polynomial derived from Schubert polynomials

Let $W=\bigcup_{n=1}^\infty S_n$ be the union of all symmetric groups $S_n$. For an element $w\in W$, denote by $\mathfrak{S}_w$ the Schubert polynomial associated to $w$, and by $\partial_w$ the ...
2
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0answers
80 views

Partial Flag Varieties and Quotients of Symmetric Polynomials

$\def\Q{\mathbf Q}\DeclareMathOperator{\Gr}{Gr}$First, consider a Grassmannian $\Gr(k, N)$ of $k$-dimensional subspaces in an $N$-dimensional space. It is known that its cohomology ring is $$H_k=\Q[...
2
votes
1answer
195 views

Noncommutative cohomology of flag varieties

Consider the Grassmannian $Gr(n, N)$ of $n$-dimensional subspaces of $\mathbf C^N$. Its cohomology ring is isomorphic to $\mathbf C[x_1, \ldots, x_n, \bar x_1, \ldots \bar x_{N-n}]/I_{n,N}$, wherethe ...
4
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1answer
128 views

Counting block-equivalent permutations

Consider the group $\mathfrak{S}_n$ of permutations on the letters $\{1,2,\dots,n\}$. We say two permutations are b-equivalent, $\pi_1\,\pmb{\sim^b}\,\pi_2$, if one can be determined from the other ...
4
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1answer
318 views

Explicit description of the Lagrangian Grassmannian as a homogeneous space

Looking at this and this question about the Lagrangian Grassmannian, and its linked Wikipedia description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit ...
3
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1answer
257 views

Lagrangian Grassmannian as a Spin Manifold

I am trying to better understand this nice answer to a question of mine, which states Spin structures on a compact complex manifold $(M^{2n},J)$ are in bijective correspondence with isomorphism ...
6
votes
1answer
291 views

Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians

The Lagrangian Grassmannian is an important example in symplectic geometry, see here or here for details. It shares many similarities with the ordinary Grassmannians (as one would expect from the name)...
3
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0answers
127 views

algebraic and topological K-theory of generalized flag manifolds

In the paper https://arxiv.org/pdf/math/9912153v1.pdf, the authors give the definition of "flag-like" varieties (Definition 5, pag. 13). They justify the choice of the name by saying (but not proving) ...
6
votes
1answer
160 views

Some intuition on the $SL_n$-module $V_{[1,1,…,1]}$

(This question highly overlaps with this and also this.) The irreducible ${\sf SL}_{n-1}$-module $V_{[1,1,\ldots,1]}$ is the one providing the minimal projective embedding $\mathbb{P}(V_{[1,1,\ldots,...
4
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0answers
114 views

Pushforwards of higher-rank vector bundles on flags

Let $V \cong \mathbb{C}^3$ and let $\pi: Fl(V) \to \mathbb{P}(V)$ be the projection from the flag variety to the projective space (of lines) of $V$. Let $L \subset H \subset \mathbb{C}^3$ be the ...
5
votes
2answers
162 views

real orbits on flag varieties

If $G$ is a complex semisimple Lie group,and $B$ is a Borel group, we can form the flag variety $G/B$. If $G_R$ is a real form of $G$, we can then let $G_R$ act of $G/B$ on the left and consider the ...
2
votes
1answer
210 views

The automorphism group of a divisor in a complete flag variety

Let $G$ be a complex semisimple Lie group, $B$ be a Borel subgroup of $G$. Denote by $X$ the quotient $G/B$. It is a complex projective variety. Let $L$ be a $G$-equivariant line bundle on $X$ such ...
6
votes
2answers
231 views

Real orbits of Complex Flag Varieties

Let $G$ be a semi-simple algebraic group over $\mathbb{C}$ with Borel subgroup $B$ and consider the flag variety $G/B$. If $G_0 \subset G$ is some real form, then $G_0$ acts on $G/B$ and decomposes ...
1
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0answers
130 views

On the maximal powers of $q$ which arise in a quantum product

Let $X=G/P$ be a generalized flag variety (where $G$ denotes a connected, simply connected, semisimple complex linear algebraic group and $P$ a parabolic subgroup). In this paper by Fulton and ...
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0answers
82 views

Rank diagrams of permutations $w \in S_{m}$ in the study of complete flag varieties [closed]

I'm looking for some good references that may either prove or help to prove the following statement: Show that a matrix $r=(r_{pq})_{1 \leq p,q \leq m}$ defines a rank diagram for some pair of ...
3
votes
1answer
220 views

Schubert varieties and Young diagrams

In his book Young Tableaux, Fulton asserts, in Exercise 9.4.18 on p. 152, that the Schubert variety $\Omega_{\lambda}$ is defined by the conditions $\text{dim}(V \cap F_{n+i- \lambda_{i}}) \geq i$ for ...
2
votes
1answer
105 views

Projections of orbifolds

A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an ...
3
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2answers
434 views

Borel--Bott--Weil for the Grassmannians

The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians? More precisely, suppose $G(\mathbf ...
4
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0answers
135 views

Mirror Symmetry for Flag Supermanifolds

I recently learned the following two things, and I wish to know how to make them reconciled. (1) As far as I understand, the flag manifolds serve as a tractable class of examples for the very ...
3
votes
0answers
291 views

Equivariant Cohomology of flag varieties

Let $G$ be a simple simply connected algebraic group and $T$ be a maximal torus in $G$. Let $B$ be a Borel containing $T$ and $N(T)/T$ be the Weyl group. We have nice actions of $T$ and $W$ on the ...
3
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0answers
120 views

How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...
6
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0answers
182 views

Mirror Symmetry for Homogeneous Spaces other than Flag Manifolds

Mirror symmetry is (reasonably) well understood for the general flag manifolds, due to the work of Kim, Givental, Rietsch, and others. Do there exist other homogeneous spaces for which mirror symmetry ...
4
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0answers
213 views

Application of Frobenius splitting in characteristic $0$

In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field. I am reading Brion and Kumar's book and I can see that there are geometric results can be ...
2
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0answers
134 views

Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$

We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space: $$ SU(n+1)/U(n) \simeq {\mathbb CP}^{n}. $$ We can split this process into two ...
2
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0answers
161 views

Schubert Calculus for the Full Flags

Almost all introductory texts on Schubert calculus discuss the Grassmannian case only. Does there exist a nice discussion of the full flag manifold case $SU(N)/T^{N-1}$? A low dimensional example ...
6
votes
1answer
201 views

Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?

Let $$ M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast} $$ be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If $P=\mathbb{...
0
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2answers
333 views

Classifying compact homogeneous Kähler manifolds

In this comprehensive answer to an old question, it is stated that Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group. ...
4
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2answers
207 views

Are Wolf spaces flag manifolds?

It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general ...
1
vote
1answer
371 views

Canonical class of partial flag variety

Let $F(d_1,d_2,\ldots,d_k;n )$ be the variety of all flags $\mathbb A^{d_1} \subset\mathbb A^{d_2}\subset\ldots\subset \mathbb A^{d_k}\subset \mathbb A^{n}$. This variety has the natural maps to ...
5
votes
2answers
636 views

Notes on flag varieties and Grassmannians for beginners

Can you suggest books or lecture notes (for beginners) covering basic material about flag varieties and Grassmannians (of reductive groups), with emphasis on the usual flag variety, i.e. flag variety ...
4
votes
1answer
209 views

Equivariant Almost Complex Structures on the Full Flag Manifolds

On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the ...
0
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1answer
162 views

The weird projection from SO(2n)/B to maximal isotropic grassmannian

Take the generalized flag variety $SO(2n,\mathbb{C})/B$, considered as the moduli of isotropic flags (according to the form $\langle e_i, e_{2n+1-j}\rangle=\delta_{ij}$) $$F_1\subset F_2\subset\cdots ...
1
vote
1answer
358 views

Geometric interpretation of Chern classes over flag manifolds

I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able ...
1
vote
1answer
138 views

Schubert Polynomials for Complex Projective Space

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific ...
2
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2answers
269 views

Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...
1
vote
1answer
182 views

Tensoring by Line Bundles to Produce Holomorphic Sections

Inspired by the line bundle case, I have the following question: Given an equivariant holomorphic vector bundle over complex projective space, is it true that tensoring it by line bundles often ...
6
votes
1answer
356 views

Quotient of Flag varieties

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, ...
5
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0answers
147 views

Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below: Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$ Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
6
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0answers
138 views

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 ...