Questions tagged [flag-varieties]
The flag-varieties tag has no usage guidance.
48
questions with no upvoted or accepted answers
17
votes
0
answers
516
views
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 ...
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
$$
\widetilde{\mathcal{N}}=T^*\mathcal{B}\rightarrow \mathcal{N}...
6
votes
0
answers
169
views
Does the $K^1$-group of a complete flag variety vanish?
For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space
$$
U(n)/T^n
$$
is called the complete flag variety of order $n$. For the special ...
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 ...
6
votes
0
answers
302
views
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 ...
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 ...
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 ...
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,-...
5
votes
0
answers
147
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
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 ...
4
votes
0
answers
135
views
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_- ...
4
votes
0
answers
170
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) ...
4
votes
0
answers
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 ...
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 ...
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 ...
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 ...
4
votes
0
answers
310
views
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\...
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).$...
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 ...
3
votes
0
answers
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}\...
3
votes
0
answers
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 ...
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 ...
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 $...
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 ...
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 ...
3
votes
0
answers
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 ...
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 $N$ ...
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, ...
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 ...
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 ...
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 ...
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 \...
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 $C_n(\mathbb{R}^d)$ ...
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 ...
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
$$H_k=\Q[...
2
votes
0
answers
161
views
Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$
We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space:
$$
SU(n+1)/U(n) \simeq {\mathbb CP}^{n}.
$$
We can split this process into two ...
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 ...
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 $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ ...
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}^{\...
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 $\mathcal B$ ...
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{\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...