Questions tagged [flag-varieties]
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143
questions
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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 ...
4
votes
2
answers
171
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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. ...
2
votes
0
answers
65
views
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 ...
3
votes
0
answers
146
views
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).$...
6
votes
2
answers
297
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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 $...
1
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1
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502
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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. ...
5
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110
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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 ...
3
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1
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59
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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 ...
2
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66
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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 ...
2
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1
answer
124
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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 ...
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4
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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.
...
5
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1
answer
211
views
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 ...
6
votes
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258
views
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 (...
14
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302
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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)...
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69
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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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341
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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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571
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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,$$
is there a classification of the cases where the LR ...
5
votes
0
answers
273
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CW-structure on flag manifolds
I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer.
Let $G$ be a compact Lie ...
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171
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Visualizing the affine Bruhat decomposition for $\operatorname{SL}_2$
$
\newcommand\Fl{\mathcal{F}\!\ell}
\newcommand\numC{\mathbb{C}}
\newcommand\numZ{\mathbb{Z}}
\newcommand\ringO{\mathbb{O}}
\newcommand\ringK{\mathbb{K}}
\newcommand\power{\...
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135
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An involution defined on the intersection of opposite Schubert cells in real full flag manifolds
Let $F_n$ denote the full flag manifold consisting of all complete flags in $\mathbb{R}^n$. For the standard basis $\{ e_1,\dots,e_n\}$ of $\mathbb{R}^n$, denote the standard opposite flags:
$\sigma_- ...
6
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169
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Does the $K^1$-group of a complete flag variety vanish?
For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space
$$
U(n)/T^n
$$
is called the complete flag variety of order $n$. For the special ...
17
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516
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Your favourite alternative proof of Borel–Weil–Bott
There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple ...
2
votes
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answers
150
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Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
20
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3k
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What is the significance that the Springer resolution is a moment map?
Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution
$$
\mu: T^*\mathcal{B}\rightarrow \mathcal{N}
$$
is the moment ...
1
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0
answers
164
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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 ...
2
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0
answers
40
views
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 $C_n(\mathbb{R}^d)$ ...
3
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2
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344
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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 ...
5
votes
1
answer
401
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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 ...
3
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0
answers
109
views
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 ...
1
vote
1
answer
365
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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 ...
14
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3
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3k
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What functor does Grassmannian represent?
As we know, the projective space $\mathbb{P}^n$ represents the functor sending $X$ to the set of line bundles $L$ on $X$ together with a surjection from the trivial vector bundle to $L$.
My question ...
5
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157
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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 ...
3
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1
answer
260
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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 ...
3
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178
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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
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1
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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{...
3
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1
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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 ...
3
votes
1
answer
193
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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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,-...
8
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1
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364
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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=\...
3
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0
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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 ...
1
vote
1
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374
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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
answer
124
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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 ...
3
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102
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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
votes
1
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232
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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 ...
3
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1
answer
237
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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 ...
3
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0
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70
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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 $...
7
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2
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2k
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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 ...
3
votes
0
answers
174
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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 ...
64
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6
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8k
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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 ...