All Questions
2,543 questions
2
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tangent bundle of the toric variety of the wonderful compactification.
Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in $\...
- 3,472
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 ...
CommunityBot
- 1
1
vote
1
answer
265
views
How we characterize a subgroup of finite group of Lie type with unipotent elements.
Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ ...
- 4,280
0
votes
1
answer
160
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subgroups of a $p$-solvable group and complete reducibility
1.
Let $G$ be a $p$-solvable group and $V$ be a finite dimensional
faithful $kG$-module, where the characteristic of $k$ is $p$. But
$V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...
- 22.6k
1
vote
0
answers
82
views
decomposition lemma in adelic groups II
Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.
Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.
On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}...
- 3,472
1
vote
0
answers
140
views
on a decomposition lemma in adelic groups
Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...
- 3,472
0
votes
1
answer
678
views
For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?
Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi ...
- 35
3
votes
2
answers
1k
views
Restriction of scalars of simple algebraic groups
I'm trying to understand the following basic property of the restriction of scalars:
Given an absolutely simple algebraic groups $G$ defined over a number field $k$, are there at most finitely (up-to ...
- 11.2k
6
votes
1
answer
666
views
Groups becoming algebraic groups
Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that:
(1) the set of $k$-points has the structure of a group.
(2) for any $g\in G$ the right-...
- 460
7
votes
1
answer
840
views
Confusing Point in Proof: Semisimple Automorphism Fixes Torus
I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof.
The setting is as follows. ...
- 52.9k
9
votes
1
answer
1k
views
Top chern class in positive characteristic
Given a nonsingular, projective variety $X$ of dimension $n$ over an algebraically closed field $k$.
Over $k=\mathbb{C}$, the top chern class $c_n(T_X)$ of the tangent sheaf is the Euler ...
- 18.7k
5
votes
0
answers
783
views
finite etale group scheme over a field
Could I have an example of a finite etale group scheme over a field k which is not a constant group scheme?
I just know that the category of etale group schemes over a k is equivalent to the category ...
- 147
12
votes
1
answer
1k
views
Recovering classical Tannaka duality from Lurie's version for geometric stacks
In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...
CommunityBot
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6
votes
2
answers
743
views
Measuring how far from being cocompact a lattice is
Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by left-...
- 3,201
1
vote
1
answer
700
views
CM liftings of abelian varieties and liftings of Frobenius
It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical ...
- 185
5
votes
1
answer
782
views
Representations of reductive groups over local fields through parahoric induction
Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...
- 6,349
1
vote
1
answer
234
views
Describing a matrix group (with integer coefficients) through conditions on the coefficients.
I'm wondering if there's always a (not too complicated?) way to characterize a matrix group by conditions on the coefficients.
I know if I'm dealing with matrix groups over a field, then it's sort of ...
- 25.2k
3
votes
1
answer
765
views
Are extensions of linear algebraic groups (over a field) themselves linear algebraic?
The title says it all.
A very similar question was asked and answered about linear groups, but none of the counterexamples are algebraic:
Are extensions of linear groups linear?
If $A$, $B$ are ...
CommunityBot
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4
votes
1
answer
1k
views
Involution of the Fermat quartic
Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by
$$x^4-y^4-z^4+w^4 = 0$$
and consider the involution
$$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$
The surface $X$ can be seen ...
CommunityBot
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3
votes
3
answers
815
views
Irreducibility of fundamental Weyl modules
It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the ...
- 52.9k
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
4
votes
1
answer
1k
views
Closed subgroups of a $p$-adic algebraic group
Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ the group of rational points of an algebraic group defined over $F$, endowed with the natural topology. Any Zariski closed subgroup $H \subset G$...
- 4,015
11
votes
2
answers
2k
views
Mostow's theorem on algebraic groups
In his classical 1956 paper
Fully reducible subgroups of algebraic groups
Mostow proves the following theorem:
Theorem 7.1.
Let $G$ be an algebraic group over a field $K$ of characteristic 0,
...
- 52.9k
2
votes
1
answer
1k
views
Are certain simple Lie groups linear algebraic groups?
Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)
Such a group should automatically ...
- 13.8k
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
CommunityBot
- 1
2
votes
3
answers
1k
views
A question on a unipotent element in reductive algebraic groups
Let G be a connected reductive group over complex numbers whose derived subgroup is simply connected. Let u be a unipotent element of G. The centralizer of u in G is denoted by Z_(u). Let F_(u) be a ...
- 11
7
votes
2
answers
1k
views
Character group of Frobenius kernels
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequently, the character ...
8
votes
2
answers
497
views
When is an orbit spherical?
I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, ...
CommunityBot
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1
vote
0
answers
269
views
how to determine the Weyl group of a diagonalizable subgroup?
Assume that $G$ is a connected compact Lie group (or a connected complex reductive group), and $K$ is a diagonalizable subgroup of $G$. It is known that the Weyl group $W_G(K)$ of $K$, defined as $N_G(...
CommunityBot
- 1
7
votes
2
answers
429
views
Separating subspaces in an irreducible representation
Suppose $G$ is a semisimple $\mathbb{R}$-algebraic group with finite center, and suppose $G$ acts irreducibly on a vector space $V$. Suppose $U \subset V$ and $W \subset V$ are subspaces.
$\mathbf{...
- 149k
5
votes
1
answer
422
views
Rational automorphisms of semisimple algebraic groups
Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...
- 2,624
3
votes
0
answers
929
views
On the structure of commutative group schemes
The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent.
I am ...
- 31
22
votes
3
answers
2k
views
Is SL(2,C)/SL(2,Z) a quasi-projective variety?
Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).
Is $SL(...
- 8,838
3
votes
1
answer
485
views
Group of connected components of the global Néron-Raynaud model of a torus
Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus.
We choose one ...
CommunityBot
- 1
2
votes
2
answers
400
views
Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups
Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)...
- 31.2k
0
votes
1
answer
375
views
For an algebraic group acting on a variety, why are orbits representable?
I suspect this is really obvious, but I'm not seeing it.
For an algebraic group $G$ acting on a variety $V$, and for a point $x \in \text{hom}(\text{Spec}(K),V)$, we define the orbit $G(x)$ to be $G(...
- 4,111
3
votes
0
answers
420
views
When is a subgroup the Weil restriction of another subgroup?
I asked this question on Math.StackExchange, to no avail. I try my chance on this one.
Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ ...
CommunityBot
- 1
7
votes
3
answers
1k
views
homomorphism into reductive groups
Let $k$ be an algebraically closed field with char($k$)$= p > 0$.
Let $P$ be a finite $p$-group. For any homomorphism
$\rho : P \rightarrow GL(n,k)$ we know that the image $im(\rho)$ can be
put ...
- 785
4
votes
0
answers
203
views
The Killing form on quantized enveloping algebras and reduction to the classical case
Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
- 3,637
6
votes
0
answers
181
views
On an interesting subalgebra of the functions on the cotangent bundle of the flag variety
Setup
Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...
- 3,637
19
votes
1
answer
1k
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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 ...
- 6,543
9
votes
3
answers
1k
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Kostant partition function: asymptotics and specifics
Let $\Phi$ denote a root system and let $\mathfrak P$ denote the associated Kostant partition function. Thus $\mathfrak P(\lambda)$ is the number of ways of writing $\lambda$ as a sum of elements of $\...
- 631
5
votes
2
answers
700
views
Why $G\to G/H$ is faithfully flat?
Some questions about algebraic groups.
Let $G$ be an affine algebraic group over algebraically closed field $k$.
Questions: Let $H$ be a closed subgroup of $G$, then (as I learnt from some paper) ...
- 778
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 38.7k
30
votes
7
answers
5k
views
Shuffle Hopf algebra: how to prove its properties in a slick way?
Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$.
We define a $k$...
- 3,738
5
votes
1
answer
712
views
Structure of abelian connected complex linear algebraic groups?
Let $G$ be an abelian connected complex linear algebraic group.
Is it true that $G$ is isomorphic to $(\mathbb{G}_m)^k\times (\mathbb{G}_a)^\ell$, where the nonnegative exponents denote repeated ...
- 14.2k
8
votes
0
answers
2k
views
Closure of an orbit under the action of an algebraic group
Setting:
Fix some field $k$. I am not very prudent about the field - although I'd prefer to assume as little as possible, you may assume as much as you want, the case of primary interest being $k=\...
- 4,902
1
vote
1
answer
666
views
Conjugacy classes in Aut(G)
Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of conjugacy classes of $G$ is described in many places.
Now, I'd like to know the structure/...
- 52.9k
3
votes
4
answers
570
views
A polynomial homomorphism from Gl to the group of units is a power of the determinant
I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism $\mathrm{Gl}...
CommunityBot
- 1
3
votes
1
answer
537
views
Points on Deligne-Lusztig varieties: Interpreting Borels in relative position as flags with conditions
Background
I am studying the paper "On the Green polynomials of classical groups" by Lusztig, in which he computes the values of the Deligne-Lusztig representation, corresponding to a Coxeter element ...
- 44.7k