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.
2,111
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Tannakian theory for Lie algebras
Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
6
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2
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Errata for Linear algebraic groups by Springer
Does any one prepared a list of errata for Linear algebraic groups by Springer.
I could not find any in Google search.
First typo that i came across is in page 6, Regular functions and ringed spaces:...
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votes
1
answer
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What does the $p$-adic closure of an arithmetic lattice look like?
Let $\Gamma$ be an arithmetic lattice in a linear algebraic $\mathbb{Q}$-group $\mathbf{G}$, that is, $\Gamma$ is a subgroup of $\mathbf{G}(\mathbb{Q})$ that is commensurable with $\mathbf{G}(\mathbb{...
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votes
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answer
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Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$
Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram
$$
\beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,.
$$
Let $P_{\{\beta_2\}}...
- 11.1k
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votes
1
answer
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Inner automorphisms of algebraic groups
I'm confused about the precise definition of an inner automorphism of an algebraic group. Here is what Milne says in his book on algebraic groups:
Let $k$ be a field, let $\overline{k}$ be an ...
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3
answers
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Suzuki and Ree groups, from the algebraic group standpoint
The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G_2(F)$ has an unusual automorphism of ...
- 3,101
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votes
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answers
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Classical reductive group schemes vs. unitary groups of separable algebras with involution --- reference request
Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...
- 2,846
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2
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Explicit description of SU(2,2)/U
Consider the real diagonal $4\times 4$ - matrix
$$I_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\...
- 13.6k
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Kernels of homomorphisms of group schemes
Let $S$ be some base scheme, $H$ a finite flat group scheme over $S$, and $\alpha: \mu_p \to H$ a homomorphism of group schemes ($p$ a prime). Is the kernel of $\alpha$ necessarily flat over $S$?
(I ...
- 36.1k
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Examples of groups for which Margulis superrigidity theorem applies
I am not an expert at all in the subject of Lie groups, lattices, arithmetic groups and rigidity. But, lately I am interested in Margulis superrigidity theorem, which in most versions can be stated as ...
- 11.1k
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1
answer
520
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Vector space objects in schemes - confusion
Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...
- 2,846
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votes
0
answers
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Why the hyperoctahedral group is a ``reductive'' group?
Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
- 52.4k
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votes
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a question on Deligne-Lusztig characters
Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be ...
- 52.4k
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Do the absolute roots restricting to a given root form a Galois orbit?
Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and ...
- 52.4k
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Integral structures via lattices
I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...
- 2,846
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votes
4
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References request: representations of classical groups
Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$?
I would like to know explicit formulas of the ...
- 3,101
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Decomposition of linear groups into free products
I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
- 2,846
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Ext group for commutative finite group schemes
EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn!
Could anyone provide a reference request about extensions of finite group schemes / Ext groups.
As far as I know the category ...
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1
answer
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$A[x]$ points of an algebraic group
Let $K$ be a field and $G$ be an algebraic group. Specifically $O(n)$ or $Sp_{2n}$. Is it true that for any ring $A$ over $K$ , $G(A)\cong G(A[x])$.
Is there any reference for such kind of results?
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(Euclidean) open orbit in an irreducible real algebraic set
Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...
- 15.3k
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Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group
Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This ...
- 933
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representability of $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$
Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a ...
CommunityBot
- 1
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votes
2
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Generating Irreducible representations of a simple lie algebra with Schur functors
Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
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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 ...
CommunityBot
- 1
2
votes
0
answers
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views
Valuations of root group elements appearing in the intersection of Iwasawa and Cartan double cosets
$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\val}{\mathit{val}}$Let $F$ be a local non-Archimedean field with valuation $\val$ and $G$ be (the
$F$-points ...
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answer
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Existence of Regular Semisimple elements of reductive groups in characteristic 0
Suppose $G$ is a connected reductive group defined over a field $F$ of characteristic $0$. Does every maximal torus contain a regular semisimple element defined over $F$?
I know that over an ...
- 8,726
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votes
3
answers
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theorem of Borel and Tits
Is there anywhere where I can read a complete proof in English of this theorem by Borel and Tits:
Suppose that $G$ is a simple algebraic group over an infinite field $k$, and that $H$ is a subgroup ...
- 2,962
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vote
1
answer
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views
Lifting Lang-Steinberg to DVR's in Characteristic 0
Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
- 3,753
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votes
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Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
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Special unitary group simply connected?
Let $C'\rightarrow C$ be a double cover of smooth projective curves over $\mathbb C$. Let $K'/K$ the corresponding function fields extension. Denote by $\tau$ the Galois involution on $K'$. Let $SU(K')...
- 55k
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votes
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answers
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views
Monoid cohomology of $\mathbb{N}$ for a linear algebraic group
Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
- 273
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Group schemes and Hyperspecial maximal compact subgroups
Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
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2
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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 & ...
- 8,157
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vote
0
answers
330
views
Borel subgroup of $Sp(4,\mathbb{C})$
I am trying to understand Borel subgroups of $Sp(4,\mathbb{C})$. I think that the following is a Borel subgroup of $Sp(4, \mathbb{C})$: the subset of $Sp(4, \mathbb{C})$ of all lower triangular ...
- 960
4
votes
1
answer
215
views
Legendre's symbol in Schrödinger model for the Weil representation
I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$.
The way to present the action of the Weil representation $\omega$ of $Sp(2n,\...
- 22.6k
10
votes
1
answer
343
views
Is Zariski closure of finitely generated matrix semigroup computable?
In general, can the Zariski closure of the semigroup of matrices $\langle M_1, \ldots, M_k \rangle$ be algorithmically computed (at least in theory)?
For this purpose I'm happy to assume the ...
- 163
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vote
0
answers
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views
What are the corner minors in $Sp(4)$?
This question relates to the question and the question.
Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 &...
- 6,121
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vote
1
answer
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A non trivial example of an anti -affine algebraic group
An anti- affine group $G$ is defined to be an algebraic group with no global sections. Examples include abelian varieties and non trivial extensions of abelian varieties by torus (in characteristic $\...
- 60.2k
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votes
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Minuscule cocharacter for reductive groups
I have a question about minuscule cocharacters, which might sound trivial to the experts:
Let $G$ be a smooth affine group scheme over $\mathbb{Z}_{p}$. Furthermore, consider a cocharacter $$\mu\...
- 36.1k
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votes
1
answer
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views
Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$
Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$.
Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a ...
- 2,003
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votes
2
answers
337
views
The Tits classes of simply connected simple real groups
Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$:
...
- 2,003
4
votes
1
answer
303
views
Possible groups appearing in a Shimura datum
Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...
- 21
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votes
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Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper
In Kottwitz's 1985 Compositio paper,
Isocrystals with additional structure, first page, paragraph 4:
Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
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7
votes
0
answers
264
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Hilbert series for invariant ring
I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with).
To put it in some broader context, let $G$ ...
- 1,253
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views
Naive question about classification of unipotent character sheaves
Let $G$ be a connected reductive algebraic group over (say) $\mathbb{C}$. The set $\hat{G}_u$ of isomorphism classes of unipotent irreducible character sheaves has some complicated classification in ...
- 5,492
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votes
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answers
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views
$G$-Invariant Differential Operators
Let $G$ be a complex algebraic group, $K$ a closed subgroup so that $X=G/K$ is a homogeneous space.
Let $\mathcal{D}(X)$ denote the algebra of differential operators on $X$. The group $G$ acts on $\...
- 1,025
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votes
0
answers
550
views
Representation theory of finite groups with additional structures
Let $H$ be a finite group, representation theory of $H$ over $\Bbb C$ essentially determines $\operatorname{Hom}(H,GL_n(\Bbb C))$ up to conjugation action of $GL_n(\Bbb C)$ for each $n$. If we replace ...
- 13.5k
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votes
0
answers
100
views
Is there a projection from $G$ to the Levi subgroup of a Parabolic subgroup?
Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel ...
- 6,121
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votes
0
answers
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views
Under what conditions are superspecial abelian surfaces isomorphic over a finite field?
Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
- 1,801
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votes
1
answer
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Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$?
Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$? Generally, let $R$ be a discrete valuation ring and $K$ its fraction field. Let $\widehat{R}$ be the completion and $\widehat{K}$ the ...
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