# Questions tagged [flag-varieties]

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120
questions

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### Relation between flag varieties of Langlands dual groups

Let $G$ be a reductive algebraic group over $\mathbb C$ and let $G^\vee$ be its Langlands dual group. What are the precise relationships between the complete flag variety $X$ of $G$ and the flag ...

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### Flag variety as monoid and Schubert calculus

The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation.
When looking ...

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### Schubert cells in G/P for reductive G

All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...

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### Are there six functors for twisted D modules?

Is there a notion of holonomic D module which admits the six functor formalism in the world of twisted D modules?
Recall that twisted D modules on $X$ are well-defined for any $T$ torsor $\tilde{X}\...

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### Homology of the free loop space of generalized flag varieties

Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{...

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### Sections of the structure sheaf of a partial flag variety on big cell

Let $G$ be a connected split reductive group over a (non-archimedean) field $K$ of char 0 with split maximal torus $T$ and a standard parabolic $P$. Denote by $W$ the Weyl group of $G$ and by $W_P$ ...

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### Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$

It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$.
For ...

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### Isomorphism between $\operatorname{End}_0(E)$ and $\operatorname{End}_0(E')$ as Lie algebra bundles

This may be a stupid question.
I'm reading this paper of Indranil Biswas, Tomas L. Gomez, V. Munoz (arXiv link). I have a problem in the proof of Theorem $5.3$.
Let $X$ be a smooth projective curve ...

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131 views

### Containment of Bruhat cells on flag variety

This question was posted at MSE, but it did not receive any answer there.
Let $G$ be a connected semisimple algebraic group over $\mathbb C$, $X$ the flag variety of $G$, $B_0$ a Borel subgroup, $\...

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### Does the $K^1$-group of a complete flag variety vanish?

For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space
$$
U(n)/T^n
$$
is called the complete flag variety of order $n$. For the special ...

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### Is there a smooth $W_{G_2}$-equivariant map from the flag manifold of $U(4)$ to that of $G_2$?

The Weyl group $W$ of $G_2$, is a group of order $12$ which is generated by the subgroup of permutations of $e_1$, $e_2$ and $e_3$ and by by the element $\tau$ which maps $(e_1,e_2,e_3)$ to $(-e_1,-...

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### Union of Schubert cells being affine

Let $k$ be a field of characteristic zero, $G$ be a reductive group with a Borel $B$ and $\mathcal{F}:=G/B$ the associated flag variety. Let $W$ be the Weyl-group of G.
Then let $S \subset W$ and $Z=\...

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154 views

### Sheaf cohomology of the complement of a schubert variety

Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...

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238 views

### Multiplicative structure for sheaf cohomology of flag varieties

Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...

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98 views

### Euler characteristic of a holomorphic homogeneous vector bundle

Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic ...

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### A “Dynkin diagram locality” property of flag varieties

For $n\ge 2$ consider the set of Plücker variables $X_{i_1,\dots,i_k}$ with $1\le k\le n-1$ and $1\le i_1<\dots<i_k\le n$ and the ring $R$ of polynomials in these variables (with complex ...

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### Flag manifolds as homogeneous Kahler manifolds

In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written
Flag manifolds exhaust all compact homogeneous Kähler ...

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171 views

### De Rham cohomology of homogeneous spaces

Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...

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181 views

### Natural vector bundle on flag variety coming from the variety of nilpotent matrices of fixed rank?

Let $F$ be a field and $V$ be a $n$-dimensional $F$-vector space, then $\{A \in End(V) | A^2 =0, \operatorname{rank} A=k \}$ gives an algerbraic variety $\mathcal{N}_{n,k}$ over $F$. There is a ...

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313 views

### On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya

I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...

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### Homogeneous space for intersection of subgroups

Suppose we have a Lie group $G$ and two subgroups $P_1$ and $P_2$. We can then study the homogeneous spaces $M_1=G/P_1$ and $M_2=G/P_2$, and bundles on these spaces associated to representations of $...

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### Relative position on flag variety

Let $G$ be a semisimple algebraic group over $\mathbb{C}$. Consider the $G$ diagonal action on $G/B \times G/B$, the orbit is indexed by $W$, the Weyl group of $G$ by Bruhat decomposition. There is a ...

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### Embeddings of flag manifolds

Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...

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### 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 $...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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,...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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[...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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)...

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### 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) ...

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### 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,...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...