All Questions
2,543 questions
5
votes
2
answers
487
views
Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$
From a result Obtained by O. Schreier and B. L. van der Waerden [Math. Sem. Univ. Hamburg 6, 303- 322 (1928)], one can show that for two fields $\mathbb F$ and $\mathbb G$, and integers $n>m>2$, ...
- 2,829
1
vote
0
answers
97
views
Degree of a commutator in a hyperalgebra or enveloping algebra
Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...
- 3,637
5
votes
2
answers
586
views
Quotient of a reductive group by a non-smooth subgroup
This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p&...
- 14.2k
1
vote
1
answer
543
views
Fixed points of group action
Let us consider the group $PGL(2,\mathbb{R})$ as the group of automorphisms of real projective line and $H\subset PGL(2,\mathbb{R})$ is a subgroup of prime order $> 2$. Is it true that there always ...
- 13
1
vote
0
answers
301
views
How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
- 26.1k
2
votes
0
answers
180
views
on geometric Satake and functions
Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field.
For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata $\overline{Gr^{\lambda}}$....
- 3,472
4
votes
1
answer
1k
views
Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,920
3
votes
1
answer
642
views
Decomposition theorem for polarized abelian varieties in positive characteristic
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
- 614
1
vote
1
answer
326
views
A Criterion for Reductivity of Lie Subgroups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...
- 4,920
5
votes
0
answers
387
views
Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?
In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...
- 5,422
2
votes
1
answer
303
views
Lifting vector fields to its resolution in char $p$
In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
- 656
1
vote
1
answer
578
views
Decomposing Semisimple Perverse Sheaves
So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do ...
- 2,902
2
votes
1
answer
533
views
S-arithmetic subgroup question
I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup:
Let $G$ be a connected ...
- 47
2
votes
1
answer
307
views
On a Strongly F-regular Pair (X, \Delta)
Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
- 165
5
votes
0
answers
253
views
Do there exist pseudo-reductive (but not reductive) groups of small dimension?
I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...
- 181
13
votes
2
answers
7k
views
Simply connected simple algebraic groups
Before asking the question I should say that I don't know much about algebraic groups and I'm not sure if the question has the right level for MO. If not, please let me know and I will delete the ...
- 569
20
votes
3
answers
3k
views
Small-index subgroups of SL(3,Z)
I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$.
The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$,
along the bottom row. I could not figure out ...
- 743
5
votes
2
answers
849
views
Stabilizers for nilpotent adjoint orbits of semisimple groups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (i.e. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...
- 4,920
1
vote
0
answers
70
views
reduced group covers of a curve
Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a ...
- 314
2
votes
0
answers
55
views
on degree zero elements in adelic groups
Let $G$ a split connected reductive group and $G(\mathbb{A})$ his points in the ring of adeles.
We have a degree map $G(\mathbb{A})\rightarrow X_{*}(Z)$ where $Z$ is the center of $G$.
Let $G(\...
- 3,472
11
votes
4
answers
3k
views
Classification of Tori of GL2, up to conjugation
Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers $m,...
- 7,722
1
vote
0
answers
203
views
Frobenius kernel for unipotent algebraic groups
Let $G$ be a connected algebraic group in positive characteristic $p$. If the Frobenius kernel $G_{(p)}=ker (F:G\to G^{(p)})$ is unipotent, do we have $G$ also unipotent?
- 97
1
vote
1
answer
509
views
Extension of unipotent algebraic groups
Let $G$ be an algebraic group with closed normal subgroup $N$. Suppose that $N$ and $G/N$ are both unipotent. Does it imply that $G$ is also unipotent?
- 97
1
vote
1
answer
376
views
on z-extensions
Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a $...
- 3,472
7
votes
0
answers
286
views
Level p characteristic 2 modular forms and thetas
BACKGROUND
Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In ...
- 5,422
0
votes
0
answers
273
views
minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,472
7
votes
2
answers
2k
views
The Lang isogeny
Let $G$ be a connected commutative algebraic group over $\mathbb{F}_q$. If $\text{Fr}_q : G \to G$ denotes the $q$-Frobenius morphism, we define the Lang isogeny $L_q$ to be the endomorphism of $G$ ...
- 3,605
10
votes
1
answer
570
views
Commutativity of the Chow ring in positive characteristic
I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$.
On p. 2, he writes the following ...
- 40.3k
3
votes
2
answers
380
views
degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$
How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.
user21574
4
votes
2
answers
1k
views
Dimension of Unipotent Radicals
A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
9
votes
2
answers
464
views
Automorphisms of $SL_n$ as a variety
What are the automorphisms of $SL_n$ as an algebraic variety?
In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $...
- 14.2k
26
votes
2
answers
5k
views
General Bruhat decomposition (with parabolic not necessarily Borel)
Here is the general Bruhat decomposition (which I have seen in various paper but never with a proof or a complete reference).
Let $G$ be a split reductive group, $T$ a split maximal torus and $B$ a ...
- 991
1
vote
0
answers
220
views
Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
3
votes
1
answer
2k
views
center of the centralizer of semisimple element
Let $G$ be an adjoint group over an algebraically closed field $k$ and $s\in G$ a semisimple element.
Let $H=C_{G}(s)^{0}$ the neutral component of the centralizer of $s$. Do we have that the center ...
- 3,472
4
votes
1
answer
998
views
Dimension of irreducible representations in characteristic p
Hi, I know that over $\mathbb{C}$ the dimension of an irreducible representation of a finite group $G$ must divide the order of the group. I've read somewhere that if $p$ does not divide the order of ...
- 407
5
votes
2
answers
399
views
Conjugation of homogeneous spaces
Let $X$ be a smooth irreducible algebraic variety
over the field of complex numbers ${\mathbb{C}}$.
Let $x\in X({\mathbb{C}})$.
Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...
- 14.2k
1
vote
1
answer
331
views
Questions about multiplicative homomorphism of $\mathbb{R}$
Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.
Question 1. Whether that $f$ is surjective implies that $f$ is injective?
...
- 435
4
votes
0
answers
472
views
Classify cross-sections of the adjoint quotient for a semisimple algebraic group?
[This question arises from trying to understand an incompletely formulated earlier question
here.]
Let $G$ be a semisimple algebraic group over an algebraically closed field $K$ of
characteristic $p \...
- 52.9k
4
votes
2
answers
1k
views
Center of the algebraic group $G_{\mathbb{R}}$ for a centerless $G$
This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of $...
- 637
1
vote
3
answers
362
views
How we obtain information about a variety from an algebraic group acting on it
Let $G$ be an algebraic group acting on a variety $V$. Which information can be obtained by looking the action of $G$, and subgroups of $G$ that fixes points of $V$?. In other words how we obtain $V$ ...
- 35
3
votes
2
answers
179
views
On local parameters at the origin in an algebraic group
Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ t_1,...
- 3,637
28
votes
3
answers
2k
views
Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
- 2,215
7
votes
2
answers
3k
views
Classification of quasi-split unitary groups
Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...
- 407
11
votes
2
answers
973
views
Rational orthogonal matrices
``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of $O(...
- 96.4k
4
votes
2
answers
1k
views
Basic question about affine group schemes
I've been reading Waterhouse's book "Introduction to affine group schemes", in part to help prepare myself for an (oral) advanced topic exam in algebraic geometry. There is one exercise in chapter 1 ...
- 43
3
votes
2
answers
721
views
Higgs bundle and stable bundle
Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle ...
- 3,472
5
votes
0
answers
266
views
On Langlands Pairing and transfer factors
In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...
- 3,472
2
votes
0
answers
574
views
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
6
votes
2
answers
562
views
Normal subgroups of $SL_2$ of a polynomial ring
What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction ...
- 11.1k