# 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.

2,143
questions

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### Extending Tannakian "dictionary" to gerbes

The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...

1
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0
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45
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### Cartan decomposition over a not-necessarily-discretely-valued field

Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...

2
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0
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85
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### What is the natural module?

Lemma 2.9 of [1]:
Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...

5
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### Classification of visible actions for *reducible* representations?

Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...

1
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0
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153
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### Computer computation of the first Galois cohomology of a $p$-adic torus?

Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus
given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$.
I want to compute, in some sense ...

6
votes

1
answer

235
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### Extensions of algebraic groups and extensions of fpqc sheaves

There are at least two ways to define $\operatorname{Ext}^p(F,G)$ when $F$ and $G$ are commutative algebraic groups over a field $k$:
Pass to the associated fppf sheaves and use an injective ...

4
votes

0
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142
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### Spaces of fixed points

I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10.
The setting ...

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278
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### Has the notion of a unipotent group scheme been studied?

The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...

3
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1
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79
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### Restriction of scalar commutes with taking maximal subtorus for semisimple group G

I was wondering such a question: for a semisimple complex Lie group $G$, whether it is true that the maximal subtorus of $\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(G)$ is $\mathrm{Res}_{\mathbb{C}/\mathbb{...

1
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0
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69
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### Decomposition of $BwBw^{-1}B$

Let $(B,N,W,S)$ be a Tits system with $W$ a finite coxeter group.
Let $w\in W$, consider $BwBw^{-1}B$, then by Bruhat decomposition, it is a disjoint union of some $BxB$, $x\in W$.
My question: Let $X\...

2
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0
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66
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### Irreducibility of Białynicki-Birula cells

Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...

3
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169
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### A property of an irreducible root system

Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...

2
votes

1
answer

262
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### Prodiscreteness of rational points of algebraic groups

Let $F$ be a field of characteristic 0 complete for a discrete non-archimedean valuation.
Let $G$ be a commutative smooth algebraic group over $F$.
Let us put on $G(F)$ the topology induced by the ...

3
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0
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49
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### Algebraicity of the group of equivariant automorphisms of an almost homogeneous variety

The base field is the field of complex numbers. Let $G$ be a connected linear algebraic group. Let $X$ be an almost homogeneous algebraic variety, i.e. $G$ acts on $X$ with a dense open orbit $U \...

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2
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### Hilbert's Satz 90 for real simply-connected groups?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=...

3
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### Smooth unipotent algebraic groups over $\mathbb A^n$

Let $G\to \mathbb A^n_{\mathbb C}$ be a smooth morphism whose fibers at any point of $\mathbb A^n$ are unipotent groups. Can we conclude that $G\simeq \mathbb A^{n+N}_{\mathbb C}$ for some $N$, as a ...

4
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1
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590
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### An algebraic group has how many representations?

Let $G$ be a connected, linear algebraic group over $\mathbb{C}$. Let $\rho:G\to \mathrm{GL}(V)$ be a rational representation. Does $\rho$ have at most countably many subrepresentations (up to ...

0
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0
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113
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### Induced action on infinitesimal thickenings by an algebraic group

Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...

3
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209
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### Action of an algebraic group $G$ on a scheme $X$ with fixed rational point

Let $X$ be an irreducible locally noetherian $k$-scheme ($k$ any field) and $G$ an algebraic $k$-group acting on $X$.
Proposition 3.1.6 in these notes by M. Brion claims
Let $a : G \times X \to X$ be ...

3
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0
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150
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### Centers and conjugacy classes of groups relative to a pair of group homomorphisms

$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...

4
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0
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189
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### Questions about the fixed point functor $X^G$ of a $G$-scheme

Let $X$ a (locally Noetherian; but not sure if that's really matter) $k$-scheme, $G$ a $k$-group scheme acting on $X$ via morphism $a:X \times G \to X$.
The fixed point functor of $X$ (where $X$ is ...

1
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1
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103
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### Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points

I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups.
Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...

2
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0
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71
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### Number of rational points of a connected reductive group in a compact subset

Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...

2
votes

1
answer

239
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### Normalizer of Levi subgroup

Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$.
Associated with this ...

2
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0
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183
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### Theorem of highest weight of semisimple Lie algebras: what fails precisely for reductive case

Let $\mathfrak{g}$ a complex semisimple Lie algebra. It is well known that by the theorem of the highest weight, all finite-dimensional complex irreps of $\mathfrak{g}$ are (up to iso) classified by ...

3
votes

1
answer

111
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### Involution of $\text{GL}_{m+n}(\mathbb{C})$ fixing Levi and exchanging parabolic subgroups

Is there any involution of $\text{GL}_{m+n}$ which is the identity on $\text{GL}_m\times\text{GL}_n\subset\text{GL}_{m+n}$ and that exchanges the positive and negative associated parabolic subgroups $...

1
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0
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127
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### Question on two types of Frobenius theorem in $p$-adic groups

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...

6
votes

1
answer

408
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### Is every complex linear algebraic group a differential Galois group?

Let $ G $ be a complex linear algebraic group. In other words, $ G $ is a subvariety of the space of $ n \times n $ complex matrices and $ G $ is a group under matrix multiplication.
Does there always ...

3
votes

1
answer

182
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### Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?

$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link
Just so everyone ...

2
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0
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141
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### What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$

In Table 22.1 on Page 193 of Malle & Testerman's book "Linear algebraic groups and finite groups of Lie type", the fixed point subgroup $G^F$ (where $F$ is a Steinberg endomorphism) of ...

5
votes

1
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159
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### When is $G(R)\rightarrow G^{\textrm{ad}}(R)$ surjective?

Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The ...

2
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1
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### Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$

$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...

7
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1
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323
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### Nilpotent orbits of a parabolic subgroup

Suppose $G$ is a reductive group over an algebraically closed field of characteristic $0$ with parabolic $P$, Levi quotient $M$, and unipotent radical $U$. We denote the nilpotent elements of $\mathrm{...

4
votes

2
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146
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### Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group).
By the Tannakian formalism, $G(k)$ can be ...

4
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0
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110
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### When is the intersection of cosets of a conjugacy class $0$-dimensional?

Let $G = \mathrm{SL}_n$ (say); let $K$ be a field. Let $g$ be a regular semisimple element of $G(K)$, and $\mathrm{Cl}_g$ its conjugacy class, considered as an algebraic variety. Then $\mathrm{Cl}_g$ ...

8
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0
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199
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### Word maps from $\mathrm{Cl}^{n+1}$ to $G^n$: quasi-injectivity?

Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same ...

5
votes

1
answer

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### Normal closure of $e_{12}$ in the congruence subgroup $\Gamma_1(p)\subset \mathrm{SL}_2(\mathbb{Z})$

$\DeclareMathOperator\SL{SL}$For an odd prime $p$, let
$$\Gamma_1(p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in \SL_2(\mathbb{Z}):\begin{pmatrix}a & b \\ c & d\end{pmatrix}\...

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0
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### Injection of $G(k)/Z(k)$ into $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...

2
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152
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### GIT quotient and orbifolds

Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...

5
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2
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324
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### Simple connectedness of Levi subgroup

Let us consider a connected and simply connected semisimple algebraic group $G$ over $\mathbb{C}$, $B$ a Borel subgroup and $T$ a maximal torus contained in $B$.
Let $P_1$, $P_2$ be two standard ...

3
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0
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90
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### $\mathbb{Z}_p$-points of a $\mathbb{Z}_p$-model of a reductive linear algebraic $\overline{\mathbb{Q}}_p$-group

Let $G$ be a (connected) reductive linear algebraic group over $\overline{\mathbb{Q}}_p$. By definition, this means that $G$ is a closed subgroup of some $\mathrm{GL}_n$. We can always find a ...

5
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1
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302
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### Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...

2
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0
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143
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### Classifying stack for finite flat group scheme

Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...

3
votes

2
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361
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### Cohomology of the partial flag variety associated with the minimal nilpotent orbit

Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a ...

3
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0
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### Arithmetic lattices are finitely presented

In the book "Kazhdan's Property (T)" by Bekka-de la Harpe-Valette, the following is stated on p.6 of the introduction:
"Of course, it is classical that arithmetic lattices are finitely ...

6
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2
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317
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### Reference for Langlands dual homomorphisms

I am looking for a reference that explains in detail the existence of Langlands dual homomorphisms. It seems that in the literature two references are given most often. The first is Borel's article ...

4
votes

1
answer

152
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### Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?

Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...

1
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0
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72
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### How to know the character table of the twisted group algebra of the symmetric group $S_4$

Given the character table of its Schur cover group, is there a way to obtain the character table of twisted group algebra from that? I am particularly interested in the symmetric group $S_4$.

13
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484
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### Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?

Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...

3
votes

0
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57
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### One parameter subgroups of reductive algebraic groups

If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...