The algebraic-groups tag has no usage guidance.

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### Tannakian Formalism

The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...

**15**

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**3**answers

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### Simple Tamagawa number calculations

As is well known, Euler proved the Basel identity $\displaystyle\sum\limits_{i=0}^{\infty} \frac{1}{n^2} = \frac{{\pi}^2}{6}$. By far the most illuminating explanation of this fact that I've seen is ...

**22**

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**2**answers

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### Definition of “finite group of Lie type”?

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...

**17**

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**1**answer

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### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

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**2**answers

359 views

### Hopf algebra of Chevalley group from the root system

Has anyone worked out a uniform way of constructing the Hopf algebra of a Chevalley group out of the root system (or, more precisely out of the root datum for reductive groups).
By "uniform", I mean ...

**17**

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**2**answers

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### When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$.
When is this function ...

**11**

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### Relation between Hecke Operator and Hecke Algebra

In the study of number theory (and in other branches of mathematics) presence of Hecke Algebra and Hecke Operator is very prominent.
One of the many ways to define the Hecke Operator $T(p)$ is in ...

**29**

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**7**answers

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### Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?
My feeling is that the answer is "yes" ...

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### Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up.
In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...

**9**

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**2**answers

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### Explicit cocycle for the central extension of the algebraic loop group G(C((t))).

Let G be a simple Lie group and let G(ℂ((t))) be its loop group.
The Lie algebra g[[t]][t-1] has a well known central extension
(see e.g.
Wikipedia) given by the cocycle
c(f,g) = Res0 < f ...

**18**

votes

**1**answer

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### Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...

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votes

**5**answers

722 views

### Non-conjugate words with the same trace

Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...

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votes

**2**answers

577 views

### Examples of non-split algebraic groups

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my ...

**5**

votes

**4**answers

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### method of finding roots of polynominal equations with arithmetic operations and roots and other functions

Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on ...

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**2**answers

366 views

### Do orbits and stable loci of group actions have natural scheme structures?

Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. ...

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### Are group schemes in Char 0 reduced? (YES)

A Theorem of Cartier (e.g. Mumford, Lecture 25) states that every separated, finite type group scheme $G/k$ over a field $k$ of characteristic $0$ is reduced. Does this result remain valid if we ...

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**1**answer

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### Are there “reasonable” criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?

Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...

**8**

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**1**answer

532 views

### Uniform proof of dimension formula for minimal special nilpotent orbit?

Given a simple Lie algebra over an algebraically closed field of good characteristic such
as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...

**8**

votes

**1**answer

807 views

### Are automorphism groups of hypersurfaces reduced ?

In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of ...

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votes

**2**answers

417 views

### Levi decomposition in disconnected linear algebraic group (characteristic 0)?

For algebraic groups or Lie groups, the subject of Levi decompositions tends to be surrounded by some mystery in the literature (and in an older question raised here). While I postpone further my ...

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**0**answers

265 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**6**

votes

**1**answer

581 views

### Chopping up Dynkin diagrams

Suppose I have a simple, simply connected (linear) algebraic group $\mathcal{G}$ over an algebraically-closed field $k$, which could have any characteristic. In fact, to keep things simple, let's ...

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**2**answers

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### $G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$

Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...

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### 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 ...

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**3**answers

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### How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?

Edit: It seems I had two different constructions mixed up in my head, namely the frame torsor and the automorphism bundle of a vector bundle. This made the main question a bit confusing. The first ...

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**2**answers

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### Equivariant normalization?

Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...

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### Failure of Jacobson Morozov in positive characteristics

The Jacobson-Morozov theorem that any nilpotent $e$ in the lie algebra of a simple algebraic group $G$ can be embedded in an $sl_2$-triple, has a restriction (in terms of the coxeter number) on the ...

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**1**answer

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### Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...

**3**

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**1**answer

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### Group Cohomology for Reductive Groups

Can anyone provide a reference to proofs of statements of the following type: The higher algebric group cohomology of a reductive group $G$ over $\mathbb{C}$ vanishes.
I am interested not just in ...

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**0**answers

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### 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, ...

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**0**answers

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### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

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**2**answers

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### Why are $S$-arithmetic groups interesting?

Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.
Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...

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### homomorphism into reductive groups

Let $k$ be an algebraically closed field with char($k$)$= p > 0$.
Let $P$ be a finite $p$-group. For any homomorphism
$\rho : P \rightarrow GL(n,k)$ we know that the image $im(\rho)$ can be
put ...

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**1**answer

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### Is every reductive group scheme etale locally trivial?

Let $S$ be a scheme over a field $k$, and let $G$ be a reductive group scheme over $S$. Let us call it trivial, if it is a pull-back of a group scheme over $k$ via the structure morphism $S\to k$. Is ...

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### Whenever I read “centraliser of maximal split torus”, I think of…

Inspired by this question
I'd like to ask something more specific:
In the theory of connected reductive groups over fields, one often reads about the centraliser of a maximal split torus. Here is ...

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votes

**2**answers

268 views

### Simple representations of products of algebraic groups

I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional.
Let $G_1$ and $G_2$ be affine algebraic group schemes ...

**3**

votes

**1**answer

539 views

### Heisenberg group in characteristic two

I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V ...

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votes

**1**answer

245 views

### Structure of abelian connected complex linear algebraic groups?

Let $G$ be an abelian connected complex linear algebraic group.
Is it true that $G$ is isomorphic to $(\mathbb{G}_m)^k\times (\mathbb{G}_a)^\ell$, where the nonnegative exponents denote repeated ...

**3**

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**0**answers

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### uniqueness of quotients of principal congruence subgroups

For each $n \geq 2$, is $\Gamma(2^{n})$ the unique normal subgroup of $\Gamma(2)$ with quotient isomorphic to $\Gamma(2) / \Gamma(2^{n})$ (here we are talking about principal congruence subgroups of ...

**3**

votes

**2**answers

278 views

### Specialisations of flag varieties

Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group.
My question concerns specialisations of flag ...

**3**

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**2**answers

338 views

### Suslin's Stability Theorem for Chevalley Groups

I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...

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**0**answers

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### Chevalley groups over $k[t]/t^n$

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and ...

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votes

**1**answer

259 views

### On the $F$-rational points of the derived group of a connected reductive algebraic group

Let $F$ be a local non-archimedean field and let $G$ be a connected reductive algebraic group defined over $F$. Let $G_{der}$ denote the algebraic derived group of $G$; this is connected and ...

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### affine schubert cells and bruhat order

Let $G$ be a simply connected group over $k=\bar{k}$, $B$ a Borel subgroup and $I$ the corresponding Iwahori in $G(k[[t]])$, $T$ a maximal torus and $K=G(k[[t]])$.
Let $\lambda\in X_{*}(T)^{+}$ be a ...

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vote

**1**answer

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### Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...

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### Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that ...

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**1**answer

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### Quotient of an algebraic group by a closed algebraic subgroup

Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" ...