All Questions
2,543 questions
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 ...
- 53
9
votes
3
answers
1k
views
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
17
votes
0
answers
585
views
Actions on ℍⁿ generated by torsion elements
Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...
- 45k
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 ...
- 359
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/...
- 6,133
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}...
- 4,902
21
votes
4
answers
3k
views
Computing the Zariski closure of a subgroup of SL(n,Z)
Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...
- 3,201
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 ...
- 4,593
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(...
- 11
3
votes
3
answers
626
views
Closed reductive sub-orbits
Let $G$ be a reductive affine algebraic $\mathbb{C}$-group (not necessarily connected). Suppose $X$ is an irreducible affine algebraic set over $\mathbb{C}$ where $G$ acts rationally. Suppose that $...
- 8,529
11
votes
2
answers
2k
views
Groups of matrices that preserve several quadratic forms
Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q_i=U Q_i U^T$, $i=1,2..,k$ What is known about such groups? (at least ...
- 2,232
3
votes
1
answer
481
views
Turing-Complete Cellular Automata and Sym(Z)
Does there exist a Turing complete, cellular automata with universe and alphabet $\mathbb{Z}$ such that the only allowable configurations are permutations of $\mathbb{Z}$? Formally, consider $\tau : ...
- 33
4
votes
1
answer
497
views
Can a non-trivial action of a connected group on a reduced scheme be trivial on a dense open?
It is well-known that if a reduced algebraic group $G$ acts on a separated reduced scheme $X$, and $G$ acts trivially on a dense open subscheme $U\subseteq X$, then the action is trivial.
If $X$ is ...
- 24.1k
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
13
votes
2
answers
1k
views
Cayley Transform for all reductive groups a.k.a an algebraic logarithm
Is it true that for every reductive algebraic $G$ over ${\mathbb C}$ with a Lie algebra $\mathfrak g$ there is an open neighborhood $U$ of the identity in $G$ and an algebraic function (in a sense of ...
- 2,390
3
votes
2
answers
675
views
The group G^+ of algebraic groups over local fields
Let $G$ be an algebraic group defined over a char 0
local field $k$. Following Borel and Tits (73) we define
the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$
generated by the unipotent elements ...
5
votes
0
answers
504
views
More familiar description of wonderful compactification of SL_n/S(GL_2 \times GL_n-2)
I am trying to learn a bit about spherical geometry and wonderful compactifications. Please correct any misconceptions. If I've understood http://www.springerlink.com/content/x62342v721707828/ ...
- 3,758
4
votes
1
answer
624
views
Quick easy question - representation theory
1) What is the proper term for a closed subgroup H of an algebraic group G such that every linear representation of H arises as the restriction of a representation of G?
2) Where can I read about ...
- 511
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
5
votes
2
answers
1k
views
Quotient space of algebraic group
Let $H \subset G$ closed subgroup of an algebraic group. We want to prove the existence of the quotient $G/H$ which is a quasi-projective variety and homogeneous G-space.
We can find a vector $0 \ne ...
- 61
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 3,362
11
votes
2
answers
2k
views
The anticanonical bundle on a flag variety is ample
Hello,
I would like to get references or answers, for the following. How do I show that the anti-canonical line bundle (i.e. dual to top wedge power of cotangent bundle) on a flag variety (of a ...
- 5,562
15
votes
3
answers
4k
views
Connectedness of the linear algebraic group SO_n
I apologize in advance if my question is too elementary for MO.
It is a well known fact that the linear algebraic group $G = \mathsf{SO}_n$ is connected, and there exist a few different proofs of ...
- 6,614
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
2
votes
1
answer
319
views
Reference request for Cartier Duality of algebraic tori
Hi,
I need a reference for the following result:
Let $S$ be a scheme and let $X$ be an algebraic torus over $S$. Then the functor $F_X :S'\mapsto Hom_{S'}(X\times S',\mathbb{G}_M\times S')$ is ...
- 527
2
votes
1
answer
236
views
Double coset isomorphism
Let $G$ be a connected reductive group (although I don't think this is relevant here), $B$ a Borel subgroup containing a maximal torus $T$ and $U$ the associated unipotent radical ($U^-$ the opposite ...
- 311
5
votes
0
answers
460
views
Has anyone used this theorem of P. Cartier?
In "Groupes Algebriques et Groupes Formels", Conf. au coll. sur la theorie des groupes algebriques, Bruxelles 1962, P. Cartier proves the following in Section 9, Theoreme 1:
(What follows is my ...
- 13.3k
7
votes
2
answers
994
views
commuting elements in a reductive group
Does anyone know if the following holds?
Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.
To make things easier, you ...
- 2,390
2
votes
2
answers
757
views
Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 33
2
votes
2
answers
636
views
Are there other ways to show Pic(G)is trivial when G is a simple-connected semisimple algebraic groups over C?
Indeed,I'm reading the book《representation theory and complex geometry》,there is a proof of the fact that Pic(G)is trivial when G is a simple-connected semisimple algebraic group over C,but the proof ...
- 73
14
votes
1
answer
1k
views
Lie groups vs. algebraic groups and deformations
I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be ...
- 1,211
1
vote
0
answers
617
views
levi subgroup generated by maximal tori?
In the levi decomposition of an connected algebraic group $G$, is the levi subgroup generated by maximal tori of $G$?.
- 11
1
vote
0
answers
116
views
Reference request: a verification of a nonstandard subgroup being a Tits subgroup.
I have a particular infinite-index subgroup $H$ of the genus 2 symplectic group $Sp(2, \mathbb{R})$. This subgroup is self-normalizing (ie. $gHg^{-1}=H$ only if $g\in H$). I am looking to determine ...
- 2,274
4
votes
2
answers
500
views
density in SU(2,1)
Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?
- 709
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
17
votes
4
answers
2k
views
What are supersingular varieties?
For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.
I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...
- 2,040
3
votes
0
answers
162
views
Does the following characterization of subgroups of $GL_2(\mathbb{F}_p)$ generalise?
Let $p$ be a prime number. By a Cartan subgroup of $GL_n(\mathbb{F}_p)$ I mean an absolutely semisimple maximal abelian subgroup.
When $n=2$, it is well-known* that, for $G \subset GL_n(\mathbb{F}_p)$...
- 2,827
3
votes
1
answer
168
views
homogenous bundles
Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I ...
- 543
26
votes
3
answers
5k
views
Questions about the Bernstein center of a $p$-adic reductive group
Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-...
- 809
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
4
votes
2
answers
579
views
Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 471
9
votes
2
answers
519
views
Are algebraic groups defined by their invariants in tensor spaces?
Let $K$ be a field of characteristic zero, and let $G \subseteq \mathrm{GL}_V$ be an algebraic group over $K$, acting faithfully on a finite dimensional vector space $V$. Let $H \subseteq \mathrm{GL}...
- 4,015
8
votes
2
answers
808
views
Lie algebras and non-smoothness of centralisers in bad characteristic
Let $G$ be a simple algebraic group over an algebraically closed
field $k$ of characteristic $p>0$. For $x\in G$, let $C_{G}(x)$
denote the centraliser, considered as a group scheme over $k$. If
$p$...
- 3,823
1
vote
1
answer
185
views
Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup.
Let G be an affine connected algebraic group, and K[G] be its coordinate ring. Let Y be a subvariety of G defined as a zero set for some f in K[G]. For which f, Y is a closed subgroup of G
- 225
3
votes
1
answer
375
views
Exact sequence of Weyl groups
If we note $A_{k}$ the category of affine algebraic groups defined over $k$ and $\mathcal{G}$ the category of finite groups, we have a functor $W:A_{k}\longrightarrow \mathcal{G}$, where $W(G)$ is the ...
- 933
2
votes
1
answer
265
views
Any local algebraic group is birationally equivalent to an algebraic group
In this paper, page $6$ the authors state the following:
By Weil’s theorem $[17]$, any local algebraic group is birationally
equivalent to an algebraic group.
Where
$[17]$ A.Weil. On ...
- 123
3
votes
0
answers
234
views
Generators and relations for the enveloping algebra of a unipotent radical
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a ...
- 3,637
4
votes
1
answer
283
views
Minimal relative Schubert modules
I am trying to better understand the definition of certain objects called minimal relative Schubert modules. My primary reference is Chapters 1 and 2 of Wilberd van der Kallen's Lectures on Frobenius ...
- 2,160