All Questions
2,543 questions
2
votes
2
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875
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Parabolic subgroups and BN-pairs
We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of $...
- 52.9k
3
votes
0
answers
289
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Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31
5
votes
2
answers
339
views
Decomposition of the ring of functions on the unipotent radical of a Borel
Background
Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be ...
CommunityBot
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4
votes
2
answers
356
views
Infinite products of representations of the additive group
Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ locally nilpotent if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. ...
- 5,654
6
votes
4
answers
921
views
Coproduct on coordinate ring of finite algebraic group
I'm reading Mukai's book "An introduction to invariants and moduli", and I am having trouble understanding one of his examples. It is example 3.49 on page 101.
The setup is as follows. Let $G$ be a ...
- 2,649
11
votes
3
answers
554
views
Uniform setting for computing orders of algebraic groups over finite quotients of the integers?
A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{...
CommunityBot
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2
votes
0
answers
577
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A question on algebraic torus
Let $T$ be an algebraic torus defined over $\mathbb Q$, $T_\infty$ be its real points and $\pi_0(T_\infty)$ be the group of connected components of $T_\infty$.
Why is the homomorphism $T(\mathbb Q)\...
- 7,902
14
votes
3
answers
2k
views
Representations in characteristic p
Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes ...
- 6,357
3
votes
3
answers
478
views
Description of $GL_3/U$
Let $U$ be the set of unipotent upper triangular matrices and $B$ the upper triangular matrices of $GL_3$. How could I describe $GL_3/U$ ? Using coordinates, in a projective or an affine space.
For ...
- 11.2k
0
votes
1
answer
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(\...
- 23.5k
1
vote
2
answers
431
views
Reductive groups over non archimedean local fields.
I want to know if connected reductive groups over non archimedean local fields have a dense countable subset. I was thinking that this should be true because if $G(\mathbb{F})$ is such group where $\...
- 8,866
3
votes
0
answers
803
views
Tamagawa number for functional fields
Let $G$ be a split semi-simple simply connected group over a global field $F$ and let
$\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well
known that $\omega$...
- 7,037
13
votes
0
answers
556
views
Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?
Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...
- 4,193
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 156k
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 656
1
vote
0
answers
189
views
Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
CommunityBot
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4
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0
answers
184
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Exotic Chains for Group Cohomology of a Complex Lie Group
Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
CommunityBot
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4
votes
0
answers
1k
views
Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
- 11.2k
10
votes
0
answers
424
views
Polynomial function from $S^3$ to $S^3$ and quaternions
I am searching the polynomial functions from $S^3$ to $S^3$.
($S^3$ is the set of vectors $x$ in $\mathbb{R}^4$ such that $\|x\|=1$)
We say $g$ is a polynomial function from $S^3$ to $S^3$, if there ...
- 663
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 $f/g$...
CommunityBot
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3
votes
1
answer
559
views
Springer isomorphisms and parabolics
Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $...
- 3,246
6
votes
2
answers
945
views
Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
CommunityBot
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2
votes
0
answers
464
views
understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
CommunityBot
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7
votes
2
answers
571
views
abelian centralizers in almost simple groups
Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite ...
- 5,654
2
votes
2
answers
571
views
What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building?
$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?
This can be phrased also as question about ...
- 52.9k
3
votes
1
answer
805
views
Finite connected groups over a perfect field of characteristic p
In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...
- 163
3
votes
0
answers
308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 8,467
12
votes
4
answers
2k
views
Finite subgroups of $PGL_2(K)$ in characteristic $p$
Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
CommunityBot
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2
votes
1
answer
302
views
finiteness of class number: a bound for semi-simple groups?
Let $F$ be a number field, and $G$ a connected semi-simple linear algebraic $F$-group, which does not contain anisotropic (simple) $F$-factors. Write $\hat{F}$ for the ring of finite adeles $F\otimes\...
- 8,866
2
votes
2
answers
765
views
Can the intersection of a maximal parabolic with a closed sub-group contain more than one maximal parabolic?
Suppose that we have a closed embedding $G_1\hookrightarrow G_2$ of reductive groups (say over $\mathbb{Q}$), and suppose that we have a maximal parabolic sub-group $P_2\subset G_2$, and a minimal ...
- 3,032
73
votes
9
answers
9k
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What are "classical groups"?
Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with ...
- 11.1k
2
votes
2
answers
544
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Reference request: The geometry of $GL_2(\mathbb{R})$ and related questions
Can anyone please recommend some good reading on the geometry of linear groups and their actions?
An example of the kind of question I am interested in: Explicitly describe a fundamental domain for ...
CommunityBot
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3
votes
1
answer
464
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Action of Non-Split Torus in Deligne-Lustzig induction
Recently I have been trying to understand Deigne-Lustzig induction in the case of $G = \text{Sl}(2,\mathbb{F}_p).$
In this case the appropriate Deligne Lustzig variety is given by $X:xy^q-y^qx = 1,$ ...
- 52.9k
1
vote
0
answers
157
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On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
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24
votes
4
answers
4k
views
Is strong approximation difficult?
Recently a colleague and I needed to use the fact that the natural map $SL_2(\mathbb{Z}) \rightarrow SL_2(\mathbb{Z}/N\mathbb{Z})$ is surjective for each $N$. I happily chugged my way through an ...
- 23k
2
votes
2
answers
839
views
Possible Borel subgroups of GL_n?
I am trying to understand the interaction between Borel subgroups of $GL_n$ and its roots. Is it correct to say that for any choice of roots among each pair of reciprocal roots
there is a Borel ...
- 23k
5
votes
0
answers
454
views
If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
CommunityBot
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7
votes
1
answer
682
views
Class number of PGL_2
Hello.
Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$.
For any finite and non empty set $S$ of valuations of $K$,
we refer to the subgroup of the adelic group $G(\...
CommunityBot
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12
votes
2
answers
688
views
reductive group orbits in P(V)?
Say $G$ is a reductive group over $\mathbb{C}$. We can take a dominant highest weight $\lambda$ and look at the action of $G$ on $X = \mathbb{P} V(\lambda)$. The stabilizer of the class of the ...
- 52.9k
7
votes
1
answer
5k
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Chevalley's Theorem on Constructible Sets
I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...
- 1,298
8
votes
1
answer
424
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Examples of exotic modules for the additive group
Let $k$ be an algebraically closed field of positive characteristic $p > 0$, and let $X$ be an intedeterminate over $k$. I am interested in the additive group scheme $\mathbb{G}_a$, that is, the ...
- 52.9k
0
votes
2
answers
386
views
Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 30.5k
5
votes
1
answer
710
views
Log resolutions on surfaces and 3-folds in characteristic p
If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
- 10.5k
1
vote
1
answer
311
views
A weird action of SL_3 on a pair of lines
Let us consider the complex projective plane $P^2$ and two distinct lines $L,L'\subset P^2$. Let us moreover consider the restriction of the natural action of $SL_3$ to $L\cup L'$. Can you tell in ...
- 23k
6
votes
3
answers
805
views
Kähler structure on a complex reductive group
Let $G$ be a complex reductive group, and $K$ a maximal compact subgroup (such that $K_{\mathbb{C}}=G$). By the polar decomposition theorem one has that, as manifolds, $G\cong T^*K$. The inherited ...
- 616
8
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0
answers
873
views
Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783
1
vote
1
answer
660
views
Centralizer of elliptic elements in $GL(2)$
Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = ...
- 6,349
7
votes
2
answers
536
views
What are the polynomial relations between these characteristic 2 "thetas" ?
Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$.
...
- 5,422
10
votes
2
answers
783
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Software for Borel-Weil-Bott in positive characteristic?
I am interested in calculating cohomology of line bundles on flag varieties $G/B$ in positive characteristic. But I really just have a bunch of scattered examples. Does there exist some kind of ...
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