Questions tagged [algebraic-groups]
Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
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Explicit example for Display Theory for p-divisible group
Recently I am studying the display theory of formal p-div groups ([1] )by Zink.
I would like to study by working on an example. As far as I understood, the display theory is a generalization of ...
- 397
4
votes
1
answer
319
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Tangent space of a finite flat group scheme
I am reconsidering an argument previously I thought obvious but now I feel I do not understand. Let $G$ be a finite flat group scheme over a finite field $k$. $G^\vee$ the Cartier dual of it. Let $TG$ ...
- 7,902
1
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0
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Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$
Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
- 60.2k
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4
answers
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exponential/logarithm for unipotent algebraic groups
Let $k$ be a field (of possibly positive characteristic), let $U_n$ denote the space of all $n \times n$ unipotent upper triangular matrices over $k$, and let $G$ be an algebraic subgroup of $U_n$ (...
- 12.1k
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1
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508
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When is the set of $n$-th powers in a group a subgroup? [closed]
Let $G$ be a non abelian group and $G_n=\{x^n | x\in G\}$ and n is integer. Is there a sufficient condition that makes $G_n$ be a subgroup of $G$ for arbitrary $n$?
- 60.2k
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1
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Reductive groups in algebraic geometry [duplicate]
In a lot of fields in algebraic geometry (e.g. GIT or topics on étale cohomology) which make use of group scheme concepts (or in more tame way of algebraic groups), the class of reductive algebraic ...
- 11.1k
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1
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Image of the Lang-Steinberg map on disconnected centralizers of semisimple elements
Let $\newcommand{\dbF}{\mathbb F}\dbF_q$ be a finite field and let $G\subseteq\mathrm{GL}_N(\bar{\dbF}_q)$ be a connected reductive group defined over $\dbF_q$. Let $F$ be the associated Frobenius map,...
- 11.4k
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2
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Representation theory of inner forms
I once heard something like "inner forms of reductive groups have the same representation theory".
Is this assertion misguided?
If this assertion is not misguided, then is there a precise ...
- 1,592
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1
answer
914
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Why is the Langlands dual group always taken over $\mathbb{C}$?
Whenever I read a statement of the Langlands conjectures for a reductive group $G$, they are formulated in terms of the Langlands dual group, which is essentially the reductive group over $\mathbb{C}$ ...
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Is this a typo in Ihara's "On discrete subgroups of the two by two projective linear group over p-adic fields"?
In Eq. (9'') on p. 227 of Ihara's paper "On discrete subgroups of the two by two projective linear group over p-adic fields" (link), where the second line says $$"\log Z_{\Gamma}(0,\chi)=1",$$ is this ...
- 60.2k
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What can the approximation of a group by some class be used for?
Recall the following concept due to Malcev and Gromov. Let $C$ be some class of groups. A group $G$ is said to be approximable by the class $C$ if for every finite symmetric subset $F\subset G$ ...
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Are the braid groups good in the sense of Toën?
In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
- 60.2k
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Is the Bruhat cell Zariski open in a connected algebraic group $G$? [closed]
Is the Bruhat cell Zariski-open in a connected algebraic group $G$?
Specifically, is the big Bruhat cell Zariski-open (and maybe Zariski-dense)?
Is it true for all the Bruhat cells?
- 60.2k
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1
answer
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How to prove that Chevalley groups over $\mathbb R$ have no compact factors
I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem.
I've been told in another thread ...
- 60.2k
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0
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Describing compact Lie groups in purely topological terms
Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, ...
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Character of a semisimple connected Lie groups [closed]
I'm trying to see why the Chevalley groups over $\mathbb C$ have no nontrivial character?
I know that a compact connected semisimple Lie group has no nontrivial character but is the compactness ...
- 52.4k
3
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1
answer
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Natural vector bundle on flag variety coming from the variety of nilpotent matrices of fixed rank?
Let $F$ be a field and $V$ be a $n$-dimensional $F$-vector space, then $\{A \in End(V) | A^2 =0, \operatorname{rank} A=k \}$ gives an algerbraic variety $\mathcal{N}_{n,k}$ over $F$. There is a ...
- 137k
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Étale endomorphism of $\operatorname{GL}_n$ surjective over an algebraic closure
I am currently reading chapter 1, exposé XXII of SGA7 and I am stuck at the following argument, left without explanation. It can be formulated like this:
Let $k$ be a separably closed field and $\bar{...
- 60.2k
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votes
1
answer
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Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups
I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
- 60.2k
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Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...
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Connected components of real Lie groups
(This is a follow-up to this question of mine.)
Is there an example of a connected reductive algebraic group $G$ over $\mathbb{R}$ such that:
$G$ is not isomorphic to a product $G_1 \times G_2$ of ...
- 13.6k
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1
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When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]
Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure
$$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$
- 291
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The number of rational semisimple conjugacy class/the Arthur-Selberg trace formula
I was trying to understand a statement in Theorem 1.5 of this where the author seems to imply that if $G$ is a reductive group over $\mathbb{Q}$ such that $G/Z(G)$ is anisotropic, then for any ...
11
votes
1
answer
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Finiteness of $H_1 \backslash G / H_2$ and the geometry of the orbits
Let $G$ be a connected reductive group over an algebraically closed field $k$. By the Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $...
- 8,394
3
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Pointwise stabilizer of an apartment of the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$
Denote by $X$ the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$. Let $\Sigma$ be the fundamental apartment of $X$. Let $\Gamma=\mathrm{SL}_n(\mathbb{Z}[\frac{1}{p}])$.
We can prove that the ...
CommunityBot
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Elliptic Maximal Tori and Elliptic Elements
I would be grateful if someone could provide a reference/proof of the following fact (or give a counterexample if I've misunderstood and it's false!)
Let $G$ be a reductive group over a field $F$ (in ...
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About the paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl
The paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl called Linear spaces with flag transitive automorphism groups (Geom. Dedicata) from 1990 annonces a very powerful ...
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361
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Quotient of an affine scheme by an étale finite group
Let $G$ be a finite étale group scheme over a field $k$ and $X=\mathrm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\mathrm{Spec}(A^H)...
- 11.4k
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Asymptotics of regular semisimple elements in finite groups of lie type
Let $G$ be a reductive algebraic group defined over $\mathbb{F}_{q}$. Let $G(\mathbb{F_{q}})$ denote the finite group of fixed points under the Frobenius map $F$. Now, we know that the set of regular ...
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What is meant by "roots in $Lie(N)$" in root space decomposition of Lie algebras?
Let $G = GL_n$ and $T$ the invertible diagonal matrices and $N$ the upper triangular matrices with only $1$'s on the diagonal. Then the Lie algebra $\mathcal{G}$ has the roots space decomposition
$$
\...
- 3,585
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1
answer
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Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group
In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...
- 137k
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$\mathrm{Sp}_n(q)$-conjugacy classes in $\mathrm{GL}_{2n}(q)$
The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in ...
- 60.2k
3
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1
answer
479
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Centraliser of regular semisimple element in $G^F$, for a connected reductive algebraic group $G$
Let $G$ be an connected reductive algebraic group over $k=\bar{\mathbb{F}_p}$. Suppose $G$ is defined over $\mathbb{F}_q$. Let $G^{F}$ be the corresponding finite group associated to $G$. Suppose $s\...
- 13.6k
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Heights of semiabelian varieties
Fix a prime number $l$. Let $K$ be a finite extension of $\mathbb{Q}$ and $R$ be the ring of integers in $K$.
In Chapter 2 of the Storrs volume (Cornell-Silverman) it is claimed that it is not ...
CommunityBot
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Extending morphisms between semiabelian varieties
In the Storrs volume (Cornell-Silverman), Chapter 2 there is Lemma 1 stating that that if you have two semiabelian varieties over a normal scheme, then a homomorphism defined over an open dense ...
CommunityBot
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What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)
Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...
- 11.4k
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$G^F$ conjugacy class of $F$-stable maximal tori, in an algebraic group $G$ defined over $\mathbb{F}_{q}$
Let $G$ be an affine algebraic group over $k=\bar{\mathbb{F}_{p}}$. Let $q$ be a power of $p$, and assume that $G$ is defined over $\mathbb{F}_q$. Let $\mathcal{T}$, be the collection of all maximal ...
- 52.4k
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Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$
Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...
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1
answer
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Decomposition of parabolic subgroup in reductive group
Let $G$ be a reductive group over $\mathbb{Q}$ and let $P_0$ minimal parabolic subgroup.
If $P_1=M_{1}N_{1} \subset P_2=M_2N_2$ are standard parabolic subgroups of $G$, then can we decompose $P_1=(...
- 52.4k
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vote
1
answer
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Relative position and change of torus
Let $G$ be a connected split reductive group over a field $k$ of characteristic $0$. Let $T$ and $T'$ be two split maximal tori of $G$ and $B \supset T, B' \supset T'$ be two Borel subgroups of $G$.
...
CommunityBot
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Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?
Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
- 99.1k
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Embed FPPF group scheme into smooth one
Let $A$ be a ring and $G$ be an affine commutative FPPF group scheme over $A$. Can we embed $G$ into a smooth group scheme over $A$?
- 3,309
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2
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732
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Plucker relations in orthogonal Grassmannian
Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $\varpi_3$. Since $\varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known ...
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Question on the proof that the Jacquet module preserves admissibility
Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\...
- 6,110
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0
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192
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Trying to understand why Eisenstein series is well defined
I am struggling to see why Eisenstein series is well defined, and I would greatly appreciate clarification.
Let
$$
E(x, \lambda) = \sum_{\delta \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) }
e^{\...
- 11.4k
4
votes
1
answer
208
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Smallest class of linear algebraic groups that is closed under intersections and contains all reductive groups
Let $k$ be an algebraically closed field. Consider a class $C$ of linear algebraic groups over $k$ such that
every reductive group is in $C$.
If $H_1 \hookrightarrow G$, $H_2 \hookrightarrow G$ such ...
- 6,158
2
votes
1
answer
194
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On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not
In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...
- 11.4k
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1
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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 ...
- 22.9k
6
votes
3
answers
1k
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Fixed points under a finite group action on projective variety
Let us have an algebraic action by a finite group G on a complex projective variety $X=\bigcup\limits_{i=1}^N X_i$, whose irreducible components $X_i$ are all smooth and of the same dimension $d$, and ...
- 1,627
2
votes
1
answer
106
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Nice Form of Vector Field
Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for ...
- 1,832