All Questions
2,543 questions
9
votes
1
answer
447
views
Are affine groups over rings of integers finitely generated?
I'll begin by saying that I'm not sure what I want to ask specifically, but pretty sure what in general, so please don't hold my misunderstandings against me, but do comment on them.
I know that the ...
- 4,593
1
vote
0
answers
409
views
Pushforward of equivariant bundles via the Frobenius morphism
Let $G$ be a semisimple algebraic group over an algebraically closed field of positive characteristic $p$ and let $B \subseteq G$ be a Borel subgroup. Set $X := G/B$, the flag variety of $G$. Also let ...
- 3,637
1
vote
2
answers
393
views
Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 16.9k
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
20
votes
0
answers
764
views
Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?
The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary,
the result of historical accidents. In order to avoid repetitions, the four infinite
families $A_\ell, B_\ell, C_\...
- 52.9k
17
votes
4
answers
4k
views
cohomology theory for algebraic groups
Is there a cohomology theory for algebraic groups which captures the variety structure and restricts to the ordinary group cohomology under certain conditions.
- 173
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
7
votes
1
answer
2k
views
Does local triviality in the fppf topology imply local triviality in the etale topology?
Given an algebraically closed field $k$, a smooth group scheme $G$ over $k$
and a principal $G$-bundle $X \rightarrow Y$, which is locally trivial in the fppf topology.
Is this bundle also locally ...
- 1,391
73
votes
9
answers
9k
views
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 ...
- 52.9k
6
votes
0
answers
411
views
Real approximation for homogeneous spaces of linear algebraic groups
Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point.
We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$.
...
- 14.2k
0
votes
0
answers
839
views
intersection of a parabolic subgroup with a subgroup
I'm interested in the following question: let $k$ be a field of characteristic zero (just for simplicity), $G$ a connected semi-simple $k$-group, $P\subsetneq G$ a parabolic $k$-subgroup, and $H\...
- 313
2
votes
1
answer
1k
views
on a characterization of parabolic subgroups
Over a base field $k$, linear $k$-groups stand for affine algebraic $k$-groups. For simplicity take $k$ to be a field of characteristic zero, as in this case one has the correspondence between ...
- 1,305
4
votes
3
answers
419
views
Is the space of polynomial functions on M_n a faithful U(gl_n)-module?
We are over some field $k$ of characteristic $0$. The general linear group $\mathrm{GL}_n$ canonically acts from the left and from the right on the space $\mathrm{M}_n$ of all $n\times n$-matrices, ...
- 33.8k
31
votes
4
answers
5k
views
The Frobenius morphism
I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...
18
votes
5
answers
2k
views
Comparing algebraic group orbits over big and small algebraically closed fields
For an affine algebraic group $G$ it's often convenient (and harmless) to work concretely over an algebraically
closed field of definition $k$ while identifying $G$ with its group of rational points ...
- 52.9k
2
votes
3
answers
1k
views
Upper bound for lowest common multiple of integers with (almost) fixed sum
For a positive integer $n$, let $f(n)$ be the maximum value of $\mathrm{LCM}(S)$ among multisets $S$ of positive integers satisfying $\sum_{i \in S} (i-1) = n$.
What is known about upper bounds for $...
- 1,500
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 ...
- 21
22
votes
3
answers
6k
views
The algebraic fundamental group of a reductive algebraic group
For a connected reductive algebraic group $G$ over a field $k$, other than the \'etale fundamental group of $G$ (regarded just as a scheme), there seems to be another notion, usually called the ...
- 4,265
7
votes
0
answers
2k
views
anisotropic and elliptic tori in GL(n)
Let $F$ be a commutative field and $n\geqslant 2$ be an integer. It is well known that the maximal anisotropic mod center tori in $G={\rm GL}(n,F)$ are of the form $T = {\rm Res}_{E/F}\; {\mathbb G}_m$...
- 6,349
3
votes
1
answer
1k
views
Richardson Classes and the Bala Carter Theorem
I am interested in trying to understand the following problem. Let $G$ be a connected simple algebraic group of type $D_n$, (with $n\geqslant 4$ even), defined over an algebraically closed field of ...
- 2,902
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 ...
- 638
2
votes
1
answer
646
views
Quotient by p-th roots of unity in characteristic p
Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\...
- 16.2k
6
votes
0
answers
413
views
Tannakian categories equivalent as abelian categories
Suppose $A = Rep_k(G)$ and $B=Rep_k(H)$ are tannakian categories and $F: A\to B$ is an equivalence of abelian categories with $F(1_A) = 1_B$ (but not a $\otimes$-equivalence). What can I say about $G$ ...
- 7,527
2
votes
2
answers
503
views
Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
- 15.4k
14
votes
4
answers
1k
views
(un)decidability in matrix groups
Given a collection of matrices $S=\{M_1, \dots, M_k\}$ in (say) $SL(n, Z), \ n>2$ does $S$ generate $SL(n, Z)?$
Similar are questions are undecidable for $n\geq 4$ (eg, given a set $S$ as above, ...
- 96.4k
2
votes
0
answers
191
views
open orbits and invariant distributions
Suppose $G$ is a p-adic algebraic group, $P=MN$ a parabolic subgroup of $G$ with its Levi decomposition, $\sigma$ be a irreducible representation of $M$, we use $I(\sigma)$ to denote the unique ...
- 2,709
2
votes
0
answers
643
views
Quotient of an algebraic group by the connected component containing identity
Suppose $G$ is a finite flat group over scheme $S$, let $G^0$ be the connected component containing identity. Is it true that the quotient sheaf $G/G^0$ is always representable by a group scheme over $...
- 1,091
6
votes
3
answers
590
views
Zariski-closed subsemigroups of SL_n(C) are groups
I would like to show that any Zariski-closed subsemigroup of $SL_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF ...
- 892
4
votes
1
answer
1k
views
Representations of reductive groups over arbitrary fields
Let $k$ be a field and $G/k$ a connected reductive group. Fix a maximal torus $T$, and let $X$ denote the group of characters of $T_{\overline k}$, where $\overline k$ is a separable closure of $k$. ...
- 41
1
vote
1
answer
692
views
Lie Group Principal Embedding
I'm reading a paper on complex semi-simple algebraic group geometry at the moment, but finding the going a bit tough since I'm missing alot of the prerequisites. Firstly, the author refers to a ...
- 1,462
2
votes
0
answers
650
views
Closed orbits and reductive groups
Let $G$ be a reductive algebraic group defined over a field $k$ and $X$ an affine $G$-variety.
In the case $k$ is algebraically closed we have the following result:
Let $x\in X$ such that the orbit $...
- 143
1
vote
0
answers
229
views
twisted forms of a given group embedded in a second group?
Consider the following question about forms of a given group that are embedded in a fixed group.
Fix for simplicity $k$ a perfect field, and $H\subsetneq G$ a pair of connected reductive $k$-groups, ...
- 1,305
11
votes
0
answers
1k
views
Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
1
vote
1
answer
296
views
Computing the connected component without primary decomposition
Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. Let $\{g_1,\ldots, g_r\}$ be a Gr"obner basis for the correpsonding ideal $\...
- 53
23
votes
2
answers
3k
views
Why are Tamagawa numbers equal to Pic/Sha?
For a connected algebraic group $G$ over a global field $K$ with adeles $A$, the Tamagawa number of $G$ is the volume of $G(A)/G(K)$. It is conjectured (and often known) to be rational, namely the ...
- 8,727
5
votes
0
answers
530
views
Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?
Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
- 3,897
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
5
votes
1
answer
1k
views
kernel of G(Z/p^2 Z)->G(Z/pZ) is the lie algebra of G over Z/pZ?
Let $G$ be an affine algebraic group defined over $\mathbf Z$. The kernel of the natural homomorphism $G(\mathbf Z/p^2\mathbf Z)\to G(\mathbf Z/p\mathbf Z)$, if abelian, is a group which comes along ...
- 5,654
6
votes
1
answer
1k
views
Decomposition of an algebraic group in an affine and a proper part
Let $K$ be a perfect field. In what follows, an algebraic group $G/K$ is by definition a group scheme of finite type over $K$.
The following seems to be well-known:
Theorem: Let $G/K$ be a ...
- 1,673
21
votes
6
answers
2k
views
How do I stop worrying about root systems and decomposition theorems (for reductive groups)?
I apologize for this being a very very vague question.
Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall ...
- 229
10
votes
0
answers
881
views
Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties
BACKGROUND:
Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting ...
- 52.9k
13
votes
1
answer
787
views
Best approximation to the Weyl group as a subgroup of a reductive group.
Let G be a reductive algebraic group over a field k. Let S be a maximal split torus, Z its centraliser and N its normaliser. The Weyl group W is then defined to be the quotient N(k)/Z(k). Now we ...
- 8,866
5
votes
1
answer
384
views
Conjugacy classes with elliptic limit points
Let $G$ be a reductive algebraic group over $\mathbb R$ and $K$ a maximal compact subgroup. Then we refer to the conjugacy class in $G$ of some $k \in K$ as an elliptic conjugacy class.
Question: ...
- 538
6
votes
1
answer
2k
views
unipotent groups, their forms and representations
For simplicity fix a base field $k$ of characteristic zero, and consider smooth affine algebraic $k$-groups. (It is understood that unipotent groups in positive characteristic are more complicated, as ...
- 313
1
vote
0
answers
418
views
Centralizers and Cartan involutions
This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.
Consider $G$ a connected non-compact semi-simple Lie group, ...
- 313
6
votes
1
answer
2k
views
how to recognize subgroups through Dynkin diagram?
Fix $\mathbb{C}$ as the base field, and reductive groups are assumed to be connected.
Consider the example $SO_N\subset SL_N$. $SO_N$ is its own normalizer in $SL_N$, and the rank is much smaller ...
- 1,305
6
votes
1
answer
644
views
question about equivariant embeddings of riemannian symmetric domains
Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...
- 1,305
3
votes
0
answers
374
views
a question about centralizers in semi-simple groups
I have a question concerning centralizers in real reductive groups. I'd like to know if the following property is available in any references.
Let $L\subset H\subset G$ be an inclusion chain of ...
- 1,305
20
votes
7
answers
9k
views
Elementary reference for algebraic groups
I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by ...
- 15.4k