For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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2
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1answer
35 views

How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
0
votes
0answers
19 views

Actions of unipotent groups

If we have an connected unipotent algebraic group $G$ over $\mathbb{F}$ (the algebraic closure of a finite field of characteristic $p>0$), with an $\mathbb{F}_q$ structure (where $\mathbb{F}_q$ is ...
0
votes
1answer
33 views

Prove that this set is a group with usual multiplication [closed]

Prove that the set of all numbers of the form: $\ P1 + P2*{\sqrt d} $ , (P1 & P2 are elements of All Quotient Numbers), (P1^2 + P^2>0) & (d is an element of All Complex Numbers) not being a ...
1
vote
0answers
59 views

The root datum of a connected algebraic group, a few questions

$T$ is a maximal torus of $G$, and $P$ is the set of characters $\beta$ of $T$ for which the weight space $$\mathfrak g_{\beta} = \{ X \in \mathfrak g : \textrm{Ad } t(X) = \beta(t)X, \textrm{ for all ...
1
vote
0answers
26 views

Why is $(\textrm{Ker } \beta)^0 = (\textrm{Ker } \alpha)^0$

Let $G$ be a connected linear algebraic group, $\beta$ a root of a maximal torus $T$ of $G$, $S = (\textrm{Ker } \beta)^0$, and assume that $Z_G(S)$ is not solvable. Then $T$ is a maximal torus of $...
2
votes
1answer
17 views

Why is $s_{\alpha}^{\wedge}(\lambda) = -\lambda$?

Let $G$ be a connected, reductive linear algebraic group with semisimple rank one. Let $H = (G,G)$, $T_1$ a maximal torus of $H$, and $T$ a maximal torus of $G$ containing $T_1$. Let $\lambda: k^{\...
0
votes
1answer
15 views

Is it always true that $N_{(G,G)}(T_1) \subseteq N_G(T)$?

Let $G$ be a connected, reductive linear algebraic group whose semisimple rank is $1$. Then $H := (G,G)$ is a connected semisimple group of rank one. Let $T_1$ be a maximal torus of $H$, and let $T$ ...
0
votes
0answers
13 views

Why are the inner forms of $GL_n(F)$ equal to $GL_m(D)$?

If $F$ is a non-Archimedean local field, then any inner form of $GL_n(F)$ is isomorphic to $GL_m(D)$, for a central $F$-division algebra of dimension $d^2, md=n$. Why is this true?
2
votes
0answers
38 views

When is Cartier dual of a finite group etale?

I am trying to solve the following exercise from Waterhouse: Introduction to affine group schemes (Chapter 6, Ex. 12 on page 53) without any success. Let $char(k)=p >0$ and let $G$ be an abelian ...
0
votes
1answer
24 views

Generic translates of a divisor intersect curves

Let $G$ be a connected algebraic group and suppose $G$ acts transitively on a proper variety $X$ (say over the complex numbers). If you pick a curve $C$ and an effective divisor $D$, I have seen a ...
0
votes
1answer
36 views

Where are these rational functions coming from?

In the proof of the theorem below (Springer, Linear Algebraic Groups), $T$ is a maximal torus of $G$, with dimension $1$, $B$ is a Borel subgroup of $G$ containing $T$, and $U$ is the set of unipotent ...
3
votes
0answers
43 views

Is quotient of open invariant subset open?

I am reading GIT book by Mumford. He needs special cases of the following conjecture several times. Conjecture Let $G$ be a reductive algebraic group acting on an irreducible affine scheme $X=Spec ...
2
votes
1answer
33 views

Is the product of algebraic groups the same as the fibre product?

Assuming we have two algebraic groups $G_1$ and $G_2$ over $k$. Then the direct product $G_1 \times G_2$ with the direct product group structure is an algebraic group. Is this the same as the fibre ...
1
vote
0answers
23 views

What are some examples of non-reductive groups?

Let $G$ be a connected linear algebraic group over an algebraically closed field. The radical $R(G)$ of $G$ is the identity component of the intersection of all Borel subgroups of $G$. We say that $...
2
votes
1answer
71 views

On the definition of a reductive group.

Wikipedia defines a reductive group $G$ as an algebraic group with trivial unipotent radical. The radical is the connected component of identity in the maximal normal solvable subgroup of $G$. The ...
2
votes
0answers
22 views

Is $\overline {G^\circ\cdot p}$ a toric variety?

Consider the algebraic torus $(\mathbb C^*)^n$. Let $G$ be a subgroup of $(\mathbb C^*)^n$ that is also a reductive group. Let $G^\circ$ be the connected component of $G$ containing the identity ...
5
votes
3answers
319 views

Why is this paragraph so short?

$G$ is a connected, reductive linear algebraic group.The reference is Springer, Linear Algebraic Groups. I am having trouble making sense out of anything in this paragraph. Proposition 7.31(ii) ...
1
vote
0answers
91 views

Does this proof (Lie-Kolchin) suffer from a loss of injectivity?

In the following proof (after "But there is a more elementary proof"), I was confused on something. Apparently we can assume without loss of generality that $V = V_{\chi}$. In this case, here is ...
2
votes
2answers
36 views

Does there exist an algebraic group $G$ such that all semisimple elements with infinite order lie in $G^0$?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. If $G$ is connected, and $1_G \neq s \in G$ is semisimple, then $s$ lies in a nontrivial torus: this is Proposition 6.4.5 ...
1
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0answers
70 views

Why doesn't $R(G)$ contain a torus?

$G$ is a connected linear algebraic group which is not solvable, and $T$ is a maximal torus of $G$, with $\textrm{Dim } T = 1$. $B$ is a Borel subgroup of $G$ containing $T$, and $U$ is the group of ...
0
votes
1answer
18 views

Why is $f(\alpha) = \frac{1}{2}\langle \alpha, f(\alpha) \rangle \alpha^{\wedge}$?

Let $(X,R,X^{\wedge},R^{\wedge})$ be a root datum (reference: 7.4, Springer Linear Algebraic Groups). Define a homomorphism $f: X \rightarrow X^{\wedge}$ by $$f(x) = \sum\limits_{\alpha \in R} \...
0
votes
0answers
15 views

number of irreducible components [duplicate]

In some cases, we can draw a variety and just count the number. For example, $xy =0$ in $k[x,y]$. Also, there are many results in algebraic geotmetry or algberaic groups, telling one how to prove the ...
2
votes
1answer
60 views

the defining polynomials of $PGL_n$ as an affine algebraic group

I have read this question. Also, there are theorems telling me that $PGL_n$, as the quotient of $GL_n$ by its center, is with no doubt an affine variety (affine algebraic group). But, is it true that ...
1
vote
0answers
27 views

$Q$ parabolic in $P$, $P$ parabolic in $G$ implies $Q$ parabolic in $G$.

I am a bit confused on the proof of this lemma. $G$ is a linear algebraic group over an algebraically closed field $k$. A closed subgroup $P$ of $G$ is called parabolic if the quotient variety $G/P$ ...
0
votes
1answer
14 views

Lie algebra stabilizes the stabilized subspaces, kills the fixed vectors

Let $G$ be a closed subgroup of $\textrm{GL}_n$, let $V = k^n$. I'm trying to show: (i): If $W$ is a subspace of $V$ stable under the action of $G$, then $W$ is also stable under the action of the ...
0
votes
0answers
9 views

A unipotent algebraic group in characteristic $0$ must be connected

I was working some exercises in Humphreys Linear Algebraic Groups and just wanted to check if my solution for this problem was correct. It seemed..too easy lol. Let $\textrm{Char } k = 0$, ...
1
vote
1answer
31 views

Non-rational G-modules

Let me recall the definition of a rational $G$-module from M. Brions notes Introduction to actions of algebraic groups (Def. 1.6) Let $G$ be an affine group scheme over $\mathbb{C}$. A rational $G$-...
0
votes
0answers
25 views

Show that $d\phi(X)f = Xf$

Let $G$ be a linear algebraic group, $V$ a finite dimensional subspace of $k[G]$ which is stable under left translation. Let $f_1, ... , f_n$ be a basis for $V$, and let $a_{ij} \in k[G]$ be regular ...
1
vote
2answers
137 views

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
0
votes
1answer
20 views

Why is $T_e \overline{\chi(G)} = \textrm{Im } d \chi$?

Let $G =\textrm{GL}_n$, $s \in G$ diagonalizable, $\sigma: G \rightarrow G$ the automorphism $x \mapsto sxs^{-1}$, and $\chi: G \rightarrow G$ the morphism of varieties $x \mapsto sxs^{-1}x^{-1} = (\...
0
votes
0answers
16 views

compute integrals on the circle group

Let $\theta(t)\in SO(2)$, where $SO(2)$ is the special orthogonal group. I want to compute $\theta(t)$ by integrating an 'angular velocity', say $\omega(t)\in\mathbb{R}$. Hence, I want to write $$ \...
0
votes
1answer
13 views

Why does conjugation by a diagonalizable matrix induce a semisimple automorphism?

This is a follow up to a question I asked earlier (Why is conjugation by a diagonal matrix a semisimple automorphism of $\textrm{GL}_n$?). Let $G = \textrm{GL}_n$, $s \in G$ a diagonalizable matrix, ...
3
votes
1answer
65 views

Why is $s_{\alpha}$ a Euclidean reflection?

Let $V$ be a finite dimensional real vector space, and $\langle-,-\rangle$ a symmetric, nondegenerate positive definite bilinear form on $V$. If $v \in V$, the Euclidean reflection about $v$ is ...
3
votes
3answers
61 views

Why are the irreducible components $T$-stable?

I'm having trouble with part of a proof (7.1.5) in Springer's Linear Algebraic Groups. Let $r: G \rightarrow \textrm{GL}(V)$ be a rational representation of a linear algebraic group $G$, $B$ a Borel ...
1
vote
0answers
25 views

Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description

What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ...
0
votes
0answers
16 views

Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator $$\textrm{Ad } : G \rightarrow \textrm{GL}(\...
2
votes
1answer
40 views

Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
0
votes
1answer
26 views

Why is $T/S$ isomorphic to $k^{\ast}$? (Remark 7.1.4 in Springer Linear Algebraic Groups)

I had a quick question about quotients of varieties. I am still not very good at them. Let $T$ be a torus, $\alpha$ a nontrivial character of $T$, and $S = (\textrm{Ker } \alpha)^0$. Since $T$ is ...
1
vote
0answers
24 views

Why does the Weyl group permute the weights?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$, and $\mathfrak g$ the Lie algebra of $G$. Then there exist characters $\chi_1, ... , \chi_t$ of $T$, and subspaces $V_i = V_{\chi_i}$ of $\...
0
votes
1answer
29 views

Torus action and multigrading.

Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? ...
2
votes
1answer
56 views

Center of a semisimple group and irreducible representations

Suppose that I am over an algebraically closed field of char $0$, and $G$ is a simply connected semisimple group. For a dominant weight $\lambda$, there is an irreducible representation $W_{\lambda}...
3
votes
1answer
67 views

reference for “wonderful compactification”

I am trying to learn about the wonderful compactification for (adjoint) semi-simple groups. Are there any good references that sketch out the full construction other than here: http://arxiv.org/abs/...
0
votes
2answers
32 views

Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular groups?...
1
vote
1answer
25 views

why does the regular action of the structure group not imply triviality of a fibre system?

Let $P$ and $X$ be algebraic varieties, $\pi:P\to X$ a morphism and $G$ an algebraic group acting on $P$. Serre calls the triple $(G,P,X)$ a fibre system an proves that if it is locally isotrivial (i....
1
vote
1answer
14 views

Problem defining morphism in Galois cohomology of algebraic group

Let $K$ be a field and $G$ an algebraic group defined over $K$. If $M\supseteq L$ are two finite Galois extensions of $K$, then the groups $\text{Gal}(M/K)$ and $\text{Gal}(L/K)$ act, respectively, on ...
0
votes
0answers
19 views

Statement made by Borel on Linear Algebraic Groups page 79

The statement: Let $K/k$ be a field extension of degree, let $V$ be a finite dimensional vector space over $K$. Let the $k-$ ification( k rational structure) be $V(k)$. Then $E$ also has ...
1
vote
1answer
29 views

Algebraic groups and restricted Lie algebras

If $G$ is an algebraic group with coordinate algebra $A=\mathcal O(G)$, say over a field $k$ of characteristic $p$, then its Lie algebra $\mathfrak g$ can be endowed with the structure of a restricted ...
1
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0answers
27 views

whats wrong with this counterexample to closed subgroups of a Torus are a torus

In Cox Little and Schenck, one result that is cited in chapter two is that if $D_n$ is the $n-dimensional$ torus, and $H < D_n$ is a closed subgroup then $H$ is itself a torus. Let the underlying ...
1
vote
1answer
38 views

If a linear transformation is defined over $F$, so is the kernel

Let $V$ be a vector space over a field $k$, and $F$ a subfield of $k$. An $F$-submodule $V_0$ of $V$ is called an $F$-structure if the natural $k$-linear map $V_0 \otimes_F k \rightarrow V$ is an ...
0
votes
1answer
38 views

On the proof that one dimensional linear algebraic groups are either isomorphic to $\mathbb{G}_m$ or $\mathbb{G}_a$.

Let $G$ be a linear algebraic group of dimension one. The proof that I am looking at, in t.a springer's book (thm 3.4.9) proceeds by showing that $G$ must be either equal to its semisimple part $G_s=\...