All Questions
2,543 questions
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315
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intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$
Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\...
- 2,274
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 16.9k
0
votes
1
answer
173
views
Show that $\mathrm{PSL}_2(C)$ is complex algebraic [closed]
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\im{im}$I meet this problem when reading Artin's book Algebra. ...
- 225
0
votes
1
answer
88
views
Isogeny of connected linear algebraic group stabilizes Borel subgroup
I am trying to understand a result on algebraic groups, namely that if $\sigma:G\to G$ is an isogeny of a connected linear algebraic group over an algebraically closed field, then $\sigma$ stabilizes ...
- 177
0
votes
1
answer
122
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Functions on products of tori
Let $T$ be an algebraic torus over an algebraically closed field $k$.
Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dotsc, n_d)$ and a function $f\in k[T]$, we can ...
- 3,472
0
votes
1
answer
378
views
basic question on quotient stacks
Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
0
votes
2
answers
321
views
Why central isogeny of reductive group over general field F map maximal F split torus onto a maximal split F torus
let $f$ be a central isogeny of reductive groups over a field F, why $f$ map a maximal split $F$ torus onto a maximal split $F$ torus.
- 43
0
votes
2
answers
132
views
What is the longest word in type $C_2$ Weyl group written in terms of a matrix?
The longest word in type $A_3$ Weyl group written as a matrix is
\begin{align}
w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & ...
- 6,211
0
votes
1
answer
117
views
Haar measure of algebraic orbits
Let $G$ be a simple linear algebraic group acting on a projective variety $X$ through rational maps. Let $x_0\in X$ with stabilizer group $H$ and assume that $G/H$ in not compact and carries a $G$-...
user1688
0
votes
1
answer
101
views
Is the toral component of a connected Lie group equal to the toral component of its radical? [closed]
Given a connected Lie group, define its toral component as the maximal connected and compact subgroup of its center.
Is the toral component of a connected Lie group equal to the toral component of ...
- 29
0
votes
2
answers
441
views
Existence of $B$-reduction of a $G$-torsor on a curve
Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup.
Given a $G$-torsor $E$ on $X$ in the ...
- 5,562
0
votes
1
answer
97
views
Automorphisms of Lie algebra of type $A_5$ modulo its center in characteristic 2
Let $L$ be classical Lie algebra of type $A_5$ over field of characteristic 2; let $M$ be the quotient $L/Z(L)$ modulo its center $Z(L)$.
What about the group of automorphisms of M?
Does anybody ...
- 11
0
votes
2
answers
523
views
Is any F-stable maximal torus contained in some F-stable Borel subgroup? [closed]
Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
- 951
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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(...
0
votes
1
answer
88
views
An explicit matrix form
In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[
\...
- 217
0
votes
1
answer
175
views
Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
- 833
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votes
1
answer
129
views
Intersection of identity components
Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
- 501
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votes
1
answer
864
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A question on standard parabolic subgroup
Let $G$ be a connected reductive group over a number field $F$ and $P_0$ its minimal parabolic subgroup. Then we call a parabolic subgroup $P$ of $G$ is standard if $P_0 \subset P$.
Let $K$ be a ...
- 1,759
0
votes
1
answer
105
views
In Type $A$, if the Bruhat graph of an element $w$ in the Weyl group is regular, then to show that $l(w)=$ # $ \{\alpha \in R^+| s_{\alpha} \le w\}$
I am trying to prove that for type $A$ , rational smoothness of Schubert varieties implies smoothness.
So suppose we are in Type $A_{n-1}$, so let $G=Sl(n,\mathbb C)$, $B=$ the group of upper ...
- 235
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votes
1
answer
181
views
Does every character from group factor through largest central subgroup?
Let $G$ be a coonected reductive algebraic group over $\mathbb{Q}$ and $A_G$ its largest $\mathbb{Q}$-split central torus over $\mathbb{Q}$.
Let $X(G)_{\mathbb{Q}}$ be the addtive group of ...
- 1,759
0
votes
1
answer
87
views
representing base changes of the unit section
Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...
- 13
0
votes
1
answer
107
views
Existence of the double coset ring on paper of Ihara
In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from $...
- 2,125
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votes
1
answer
413
views
generalization of highest weight theorem for semisimple lie algebras
Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...
- 853
0
votes
1
answer
173
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Reductive subgroup and its derived subgroup with an irreducible represenation
Could you please answer the following question: Let $V$ be a faithful irreducible representation of a connected reductive group $H$ defined over $\mathbb{R}$ Is it true that the derived group of $H$, ...
- 601
0
votes
1
answer
791
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Maximal subgroups of indefinite special orthogonal group SO(p,q)
Can someone answer the following question:
Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions:
...
- 601
0
votes
1
answer
63
views
Reference for finite number of Weyl groups of reductive groups of rank $r$
I'm posting this question on behalf of someone without access to mathoverflow:
Can anybody give me a reliable reference (not a proof) to the following statement?
Up to isomorphism, there are only ...
- 3,362
0
votes
1
answer
182
views
Tables of data associated to reductive algebraic groups?
I am looking for a reference that contains lots of calculations for specific examples of various objects one can associate to a reductive algebraic group. For example, given a (specific) linear ...
- 11
0
votes
1
answer
160
views
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 ...
- 491
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
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votes
0
answers
84
views
is a linear algebraic group over an extension of $\mathbb{Q}_p$ a locally pro finite group?
Let $F$ be a non archimedean local field and let $G$ be linear algebraic group over $F$. I do not have a lot of experience with linear algebraic group, but it seems very obvious that $G$ inherits the ...
- 367
0
votes
0
answers
118
views
Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
- 6,006
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votes
0
answers
62
views
Involutions in $\operatorname {PSO}(4,K)$
In the algebraic group $G=\operatorname {PSO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there and what are the ...
- 217
0
votes
1
answer
102
views
An explicit matrix form in the symplectic group
In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[...
- 217
0
votes
0
answers
64
views
What does it mean for a linear algebraic group to act reductively
I was reading this paper by Baues and on page 918 he mention that $S$ acts reductively on the cochain complex and on page 919 again he mention the word "Since $T$ acts reductively on the complex.....
- 191
0
votes
0
answers
246
views
What does the set of all fundamental coweights look like?
Let $\Phi$ be an irreducible root system in a Euclidean vector space $V$. Let $W$ denote its Weyl group. Choose a base $\Delta=\{\alpha_1,...,\alpha_r\}$ for $\Phi$. Then $\Delta$ is a basis for $V$. ...
- 2,751
0
votes
0
answers
124
views
Krull dimension of ring of invariants
Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
- 2,907
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votes
0
answers
134
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Bound on the number of connected components of a linear algebraic group $G<\mathrm{SL}_n$?
Let $G<\mathrm{SL}_n$ be a linear algebraic group defined over a field. Is there a bound on the number of connected components of $G$ in terms of $n$ alone?
(The bound will evidently not be any ...
- 20.2k
0
votes
0
answers
71
views
Integrating homomorphisms of Borel subalgebras
Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
0
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0
answers
220
views
The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
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votes
0
answers
382
views
Theory of group representation for compact groups
I write here because some experts could help on that. It is very well known (at least for me) many reference books on linear representations of finite groups (for instance, the very classical and ...
- 1,561
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votes
0
answers
167
views
Extension of action in algebraic group
I was asking this on stack exchange but I didn't get the answer.
Borel's book Linear Algebraic Groups contains the following result
10.9 Theorem. Let $G$ be a connected affine group of dimension one. ...
- 271
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0
answers
125
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Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?
Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
- 617
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0
answers
134
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How can Borel-de Siebenthal theory be generalized?
Borel-de Siebenthal theory can be thought of as an algorithm that, given a semisimple compact Lie group $G$, gives all semisimple compact Lie subgroups whose root systems have the same rank as $G$’s.
...
- 2,773
0
votes
0
answers
192
views
algebraic subgroups of $\mathbb{G}_m$
Let $k$ be a field. How do you prove that the algebraic subgroups of $\mathbb{G}_m=\mathrm{Spec}\,k[x, x^{-1}]$ are $\mathbb{G}_m$ itself and the roots of unity $\{x^n=1\}$ for some $n$?
- 1
0
votes
0
answers
55
views
Product of subsets of a unipotent group
Let $N$ be a unipotent algebraic group and $X,Y$ be two algebraic subsets of $N$.
It is known that if $X,Y$ are algebraic subgroups of $N$, then the product
$X\cdot Y$ is closed (algebraic subset).
My ...
- 309
0
votes
1
answer
232
views
Definition of group scheme [closed]
Consider the definition of group scheme in Stack Project [022R]. In the paragraph following definition 39.4.1, it is said that
We have morphisms of schemes over $S$: (identity) $e:S\rightarrow G$ and ...
- 1,168
0
votes
0
answers
99
views
Unimodular matrices fixing $(1, 1, \cdots, 1)$
What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
- 376
0
votes
0
answers
115
views
Equivalence between coactions and actions plus a linearization line bundle
Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all ...
- 565
0
votes
0
answers
128
views
How to find the polynomials that define a compact Matrix Lie group from its Lie algebra?
Consider a compact (connected) Lie group, or more generally, a linear algebraic Lie group. Suppose we are given the Lie algebra corresponding to the Lie group. How can we find a set of polynomial ...
- 1