Questions tagged [flag-varieties]

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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 ...
7 votes
0 answers
163 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 ...
Zhaoting Wei's user avatar
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7 votes
0 answers
888 views

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}...
Zhaoting Wei's user avatar
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6 votes
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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 ...
Quin Appleby's user avatar
6 votes
0 answers
451 views

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 ...
Jan Weidner's user avatar
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6 votes
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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 ...
Jean Delinez's user avatar
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5 votes
0 answers
273 views

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 ...
Lennart Meier's user avatar
5 votes
0 answers
157 views

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 ...
Qixian Zhao's user avatar
5 votes
0 answers
108 views

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,-...
Malkoun's user avatar
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5 votes
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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 ...
Han Jin Ma's user avatar
5 votes
0 answers
169 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 ...
Qiao's user avatar
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4 votes
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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_- ...
sunny's user avatar
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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) ...
user543627's user avatar
4 votes
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159 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 ...
Jake Levinson's user avatar
4 votes
0 answers
159 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 ...
Jaak van der Smut's user avatar
4 votes
0 answers
168 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 ...
Han Jin Ma's user avatar
4 votes
0 answers
269 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 ...
Ben's user avatar
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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\...
Zhaoting Wei's user avatar
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3 votes
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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).$...
Igor Makhlin's user avatar
  • 3,493
3 votes
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 ...
F.H.A's user avatar
  • 201
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178 views

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}\...
Pulcinella's user avatar
  • 5,506
3 votes
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182 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 ...
KKD's user avatar
  • 463
3 votes
0 answers
102 views

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 ...
Igor Makhlin's user avatar
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3 votes
0 answers
70 views

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 $...
Peter Kravchuk's user avatar
3 votes
0 answers
174 views

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 ...
userabc's user avatar
  • 667
3 votes
0 answers
443 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 ...
Jack's user avatar
  • 31
3 votes
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255 views

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 ...
Zhaoting Wei's user avatar
  • 8,657
3 votes
0 answers
165 views

(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 $N$ ...
George Melvin's user avatar
3 votes
0 answers
439 views

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, ...
George Melvin's user avatar
3 votes
0 answers
160 views

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 ...
user26795's user avatar
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 ...
Qixian Zhao's user avatar
2 votes
0 answers
66 views

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 ...
EJB's user avatar
  • 153
2 votes
0 answers
150 views

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 \...
Tommaso Scognamiglio's user avatar
2 votes
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 $C_n(\mathbb{R}^d)$ ...
Malkoun's user avatar
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2 votes
0 answers
29 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 ...
Dirk's user avatar
  • 809
2 votes
0 answers
112 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[...
Bubaya's user avatar
  • 259
2 votes
0 answers
161 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 ...
Ago Szekeres's user avatar
2 votes
0 answers
196 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 ...
Han Jin Ma's user avatar
2 votes
0 answers
164 views

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 $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ ...
batconjurer's user avatar
2 votes
0 answers
382 views

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}^{\...
Johan's user avatar
  • 213
1 vote
0 answers
69 views

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 $\mathcal B$ ...
IntegrableSystemsEnthusiast's user avatar
1 vote
0 answers
171 views

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{\...
Gaussler's user avatar
  • 295
1 vote
0 answers
164 views

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 ...
KKD's user avatar
  • 463
1 vote
0 answers
124 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 ...
Lars Pettersen's user avatar
1 vote
0 answers
141 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 ...
Christoph Mark's user avatar
1 vote
0 answers
369 views

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 ...
Balerion_the_black's user avatar
0 votes
0 answers
102 views

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 ...
FreddyG's user avatar
  • 11
0 votes
0 answers
245 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 ...
Lars Pettersen's user avatar