All Questions
2,543 questions
6
votes
0
answers
402
views
What is known about line bundles on the tangent bundle of a flag variety?
Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. (I'm most interested in the positive characteristic case). Let $B \subseteq G$ be a Borel ...
- 3,637
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 3,625
5
votes
2
answers
3k
views
When does an irreducible representation remain irreducible after restriction to a semi-simple subgroup?
I suppose this question is probably elementary for experts, but I'd like to present my arguments, about which I have some doubts, and see if they are correct, or if corrections and improvements are ...
- 1,305
10
votes
1
answer
966
views
Littlewood-Richardson rule and commutativity morphism
Background
Irreducible finite dimensional representations
of the group $GL_n$ are parameterized by the highest weights,
that is by nonincreasing sequences of integers
$$
\lambda_1 \ge \lambda_2 \ge \...
- 39.3k
3
votes
1
answer
1k
views
Quotients of unipotent groups
Let $U (\mathbf{R})$ be the standard unipotent subgroup of $SL(3, \mathbf{R})$. So $U(\mathbf{R})$ is the group of 3 by 3 upper triangular matrices with 1s on the diagonal. I am interested in the ...
- 751
4
votes
1
answer
848
views
Uniform Quotient vs Universal Quotient
What is a quotient of an affine scheme that is not a universal quotient? Let's recall some terminology.
Suppose that $k$ is an algebraically closed field and $G$ is a reductive group acting on an ...
- 3,284
3
votes
1
answer
476
views
What is a closed orbit that is not separable? (ANSWERED)
What is an example of an action of a linearly reductive group variety acting on an affine variety with the property that there exists a closed orbit that is not separable?
To be more precisely, let's ...
- 3,284
8
votes
4
answers
3k
views
"Why" is every polynomial representation of SL(2) selfdual?
Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...
- 33.8k
1
vote
0
answers
186
views
Duflot-type theorem for Hopf algebras ?
In group cohomology Duflot's theorem states that the depth of the mod p cohomology ring of a finite group is greater than or equal to the p-rank of the center of a Sylow p-subgroup.
Is there a ...
- 11
4
votes
1
answer
677
views
An identity for sheaf cohomology of flag varieties
Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection.
Given a representation ($\theta,V$) ...
user6311
3
votes
2
answers
452
views
Is there an invariant theory explanation of the orbit structure of GL₂ acting on second-diagonal symmetric matrices by g∙X = gXJg^tJ ?
Statement of the Specific Result
Let $J$ denote the matrices with ones on the "second diagonal", meaning the diagonal between the (1,n) and (n,1) entry, and zeros elsewhere. So in the case $n=2$, ...
5
votes
0
answers
413
views
Frobenius splitting of tangent bundles of flag varieties
BACKGROUND
Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X \to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity ...
- 3,637
2
votes
1
answer
228
views
Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?
Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?
see also the link:mathoverflow.net/questions/36762,
- 709
9
votes
6
answers
3k
views
Explicit equations for Schubert varieties
How can one compute the Schubert variety (by compute I mean having actual polynomials that define it) for SL(n)? If this is well known forgive my ignorance and just point me to the right book/paper.
...
- 751
5
votes
2
answers
680
views
Finite group scheme acting on a scheme such that there is an orbit NOT contained in an open affine.
In Mumfords book on abelian varieties there is a theorem (on page 111) whose hypothesis is "Let G be a finite group scheme acting on a scheme X such that the orbit of any point is contained in an ...
- 467
13
votes
4
answers
5k
views
Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
- 14.8k
11
votes
2
answers
1k
views
Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup
Let $G$ be a simple Lie group over ${\mathbb C}$, $P \subset G$ a parabolic subgroup, and $L \subset P$ its Levi subgroup. Let $\lambda$ be a $G$-dominant weight and $V_G^\lambda$ an irreducible ...
- 39.3k
16
votes
1
answer
5k
views
Grothendieck-Messing theory for finite flat group schemes
Classical Grothendieck-Messing theory relates deformations of $p$-divisible groups to lifts of the Hodge filtration (if the ideal defining the nilpotent immersion is equipped with a PD-structure). If ...
- 21.3k
4
votes
0
answers
571
views
Étale cohomology of linear groups
This is in a sense a follow up question to the answer here Analytic tools in algebraic geometry
Let $k$ be an algebraically closed field of positive characteristic and let $R$ be the result of ...
- 23.5k
15
votes
0
answers
885
views
How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?
This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 (...
- 52.9k
25
votes
3
answers
2k
views
Suzuki and Ree groups, from the algebraic group standpoint
The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G_2(F)$ has an unusual automorphism of ...
- 13.3k
1
vote
1
answer
434
views
Tori acting on vector spaces
Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now ...
- 11.2k
8
votes
3
answers
2k
views
theorem of Borel and Tits
Is there anywhere where I can read a complete proof in English of this theorem by Borel and Tits:
Suppose that $G$ is a simple algebraic group over an infinite field $k$, and that $H$ is a subgroup ...
- 115
14
votes
2
answers
1k
views
Can a reductive group act non-linearly on a vector group?
Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$.
1. Some motivation
A vector group is an ...
- 3,246
4
votes
1
answer
1k
views
Group Cohomology for Reductive Groups
Can anyone provide a reference to proofs of statements of the following type: The higher algebric group cohomology of a reductive group $G$ over $\mathbb{C}$ vanishes.
I am interested not just in ...
- 1,180
5
votes
2
answers
462
views
Left U_n-invariants of SL_n - an exercise in Kraft-Procesi
I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S_n$, and I don't want these to get ...
- 33.8k
1
vote
1
answer
492
views
Rational points
Let $G$ be an affine algebraic group defined over a field of characteristic zero $K$. Suppose $G$ has only one single $K$-point, can we conclude that $G$ does not have more points?
- 143
10
votes
2
answers
1k
views
Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"?
At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
- 54.6k
15
votes
2
answers
4k
views
Hopf algebra duality and algebraic groups
Background:
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be ...
- 3,637
7
votes
3
answers
1k
views
Applications of non-reductive GIT
Geometric invariant theory works well when the algebraic group $G$ acting on a variety is reductive. There has been recent work by Doran and Kirwan here and here to find a canonical method of ...
- 508
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
- 5,422
10
votes
1
answer
412
views
Reference for Pic(G) and central extensions.
Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr_1$, $pr_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$.
Then I'm ...
- 13.3k
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
22
votes
3
answers
5k
views
Do semisimple algebraic groups always have faithful irreducible representations?
For simplicity, I will be talking only about connected groups over an algebraically closed field of characteristic zero.
The basic theorem of affine algebraic groups is that they all admit faithful, ...
- 7,273
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
5
votes
2
answers
1k
views
Conjugate cocharacters in a maximal torus
Let $G$ be a linear algebraic group over an algebraically closed field $k$, and $T$ a maximal torus of $G$.
Suppose we have two cocharacter $\mu, \mu' : \mathbb{G}_m \to T$, which are conjugate under ...
- 1,500
3
votes
4
answers
2k
views
Simplicity of (complex) orthogonal groups
I need a reference for the proof that the complex orthogonal group
$SO_{2n+1}($ℂ$) = \{A\in SL_{2n+1}($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true).
How about ...
- 331
10
votes
0
answers
465
views
A uniform bound for a "true" non-congruence subgroup
Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...
- 638
27
votes
1
answer
3k
views
Definitions of real reductive groups
There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose.
The set ...
- 971
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 27.9k
9
votes
4
answers
1k
views
Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields
My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be ...
- 638
10
votes
4
answers
1k
views
Algebraicity of holomorphic representations of a semisimple complex linear algebraic group
Let $G$ be a complex linear algebraic group, given to us as a closed subgroup of some $\mathrm{GL}(n,\mathbb{C})$. Suppose moreover that $G$ is semisimple. Then it's a fact that every finite-...
- 2,713
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
10
votes
3
answers
2k
views
How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?
Edit: It seems I had two different constructions mixed up in my head, namely the frame torsor and the automorphism bundle of a vector bundle. This made the main question a bit confusing. The first ...
- 45.7k
1
vote
2
answers
2k
views
The normalizer of a reductive subgroup
Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the ...
- 4,280
0
votes
0
answers
700
views
Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
- 2,709
5
votes
1
answer
499
views
software for computations on flag varieties in arbitrary characteristic
Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...
- 5,846
6
votes
2
answers
597
views
Points of reductive groups
Let $G$ be a (connected) reductive group over a field $k$. Then there is a natural functor from the category of representations of $G$ to the category of representations of $G(k)$.
Under which ...
- 1,209
2
votes
1
answer
474
views
Automorphism of algebraic group preserving a hyperspecial maximal compact
Suppose that $K/\mathbb{Q}_l$ is a finite extension, with ring of integers $\mathcal{O}_K$. Suppose $\mathcal{G}/K$ is a (linear) algebraic group (connected+reductive), and $\Gamma\subset \mathcal{G}(...
- 1,233