Questions tagged [flag-varieties]
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151 questions
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Equivariant Künneth formula for partial flag variety
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...
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Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
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131
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Is this family of varieties "well known"?
In my research, one can find as a special case the following family of varieties. Fix integers $0<k<n$ and let $G=Gr(k,n)$ be the Grassmannian of $k$-planes in an $n$-dimensional vector space $V$...
- 495
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$\mathbb P^1$-bundle on a partial flag variety
Let $X$ be the partial flag variety of flags $0 \subset V_k \subset V_{k+2} \subset V$ where $V$ is a fixed vector space of dimension $n$ and ${\rm dim} V_k = k$ and ${\rm dim} V_{k+2} = k+2$. Is it ...
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Can one talk abstractly about the linear independence of points on the symmetric product of copies of a flag variety?
If one considers the (complete) flag variety $F(\mathbb{C}^2)$ of $\mathbb{C}^2$, then this is biholomorphic to $\mathbb{P}^1_\mathbb{C}$, and thus the symmetric product of $d$ copies of $F(\mathbb{C}^...
- 5,215
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113
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Smoothness of some varieties related to the Slodowy slice
Let $G$ be a complex algebraic group with simple Lie algebra $\mathfrak{g} = \operatorname{Lie} G$. Let $\mathcal{B}$ be the flag variety consisting of all the Borel subalgebras of $\mathfrak{g}$.
Let ...
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155
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Schubert varieties and cohomology vanishing
To fix (albeit standard) notation, let $G$ be a complex semisimple algebraic group, and $T \subset B \subset G$ choices of maximal torus and Borel subgroup, respectively. Let $X^\ast(T)$ be the ...
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Explicit formula for complex structure on flag manifold/isospectral matrices?
Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
- 637
6
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392
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Parabolic subgroups of reductive group as stabilizers of flags
$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
- 507
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References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
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118
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Different definitions of the thick affine flag variety
I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same.
Some ...
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166
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Defining ideal of a Schubert variety as a kernel
Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n).$...
- 3,513
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geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety
For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For ...
- 185
3
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71
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Alternative bases of symmetric polynomials in cohomology ring of flag varieties and coinvariant algebras
$\DeclareMathOperator\Fl{Fl}$It is known that $H^*(\Fl(m)) \cong R^{\mathbb Z}(m)$, where $\Fl(m)$ denotes the variety of complete flags in $\mathbb C^m$, and $R^{\mathbb Z}(m)$ is the coinvariant ...
- 185
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99
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Relative position of Borel subgroups for the symplectic group
Background
Let $n$ be a positive integer, let $W$ be the Weyl group of $\text{GL}_n$.
Its set of Borel subgroups is isomorphic to the full flag variety $\mathcal{F}_n$.
In this question, I studied ...
- 153
5
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222
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Commuting matrices and cyclic modules
Let $A, B\in M_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a ...
- 3,472
7
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386
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Peterson's quantum cohomology of G/P lectures
Dale Peterson famously gave a series of lectures on the quantum cohomology of flag varieties $G/P$ at MIT in 1997. These lectures are often cited in subsequent papers by other authors on the subject (...
- 550
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81
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Polynomiality of the equivariant Euler characteristic of a sheaf tensored with a standard line bundle on the flag variety
Let ${\mathcal B}=Fl(V)$ be the variety of complete flags in an $(m+2n)$-dimensional vector space $V$ over $\mathbb C$. Then we have the standard line bundles $\mathcal{O}(\lambda)$ on $\mathcal B$ ...
4
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393
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Frobenius pushforward of an equivariant tautological bundle on the flag variety
Let $m,n$ be nonnegative integers. Let $k$ be a field of positive characteristic. Let the vector space $V$ over the field $k$ have basis $e_1,...,e_{m+n}, f_1,..., f_n$. Let $k^*$ act on $V$ by $t \...
8
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586
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When the Littlewood-Richardson rule gives only irreducibles?
Given the famous Littlewood-Richardson rule, in terms of Schur polynomials:
$$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$
is there a classification of the cases where the LR ...
5
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345
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CW-structure on flag manifolds
I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer.
Let $G$ be a compact Lie ...
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216
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Visualizing the affine Bruhat decomposition for $\operatorname{SL}_2$
$
\newcommand\Fl{\mathcal{F}\!\ell}
\newcommand\numC{\mathbb{C}}
\newcommand\numZ{\mathbb{Z}}
\newcommand\ringO{\mathbb{O}}
\newcommand\ringK{\mathbb{K}}
\newcommand\power{\...
- 295
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144
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An involution defined on the intersection of opposite Schubert cells in real full flag manifolds
Let $F_n$ denote the full flag manifold consisting of all complete flags in $\mathbb{R}^n$. For the standard basis $\{ e_1,\dots,e_n\}$ of $\mathbb{R}^n$, denote the standard opposite flags:
$\sigma_- ...
- 41
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1
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360
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Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups
If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here)...
- 32.4k
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597
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Your favourite alternative proof of Borel–Weil–Bott
There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple ...
2
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187
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Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
- 1,061
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177
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Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $\mathbb{P}^1$
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
- 473
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41
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On the higher-dimensional Berry-Robbins problem
Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...
- 5,215
3
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396
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Character which defines canonical bundle on flag variety
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half ...
- 473
5
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437
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Tensor product of perverse sheaves on flag varieties
I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example - consider the product of the big projective on $\mathbb{P}^1$ with itself (This is ...
- 700
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117
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Fundamental representation bases and generalized minors
Let $G$ be a simple simpy connected complex algebraic group.
I was wondering if there is a clear relationship between the generalized minors (defined by Berenstein, Fomin and Zelevinsky) and bases of ...
- 201
5
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180
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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 ...
- 403
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105
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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 ...
- 11
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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 ...
- 1,677
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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}\...
- 5,701
5
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1
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187
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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{...
- 1,677
3
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137
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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 ...
- 1,343
2
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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 the paper "Automorphisms of moduli spaces of vector bundles over a curve" of Indranil Biswas, Tomas L. Gomez, V. Munoz (arXiv link). I have a ...
- 297
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1
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209
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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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170
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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 ...
- 141
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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,-...
- 5,215
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398
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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=\...
- 473
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188
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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 ...
- 473
1
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1
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413
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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 ...
3
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1
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138
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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 ...
- 643
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104
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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 ...
- 3,513
4
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1
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252
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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 ...
1
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1
answer
408
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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 ...
3
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1
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242
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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 ...
- 6,552
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506
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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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