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
votes
4answers
59 views

Index of $\langle a^4\rangle$ in the group $\langle a\rangle$

Let $a$ be an element of order 30 in a group $G$. What is the index of $\langle a^4\rangle$ in the group $\langle a \rangle$? The answer is 2, however I have no idea on how to obtain that answer. ...
1
vote
2answers
50 views

Does the quotient of an algebraic group by its neutral component always split?

Let $G$ be a complex algebraic group. Let $G^\circ\subseteq G$ be the connected component of $G$ which contains the neutral element $1\in G$. Then, it can be shown that $G^\circ\trianglelefteq G$ is a ...
1
vote
1answer
64 views

A question about reductive groups's structure.

There is a flaw somewhere in the following argument but I can't track it. Take a reductive connected affine algebraic group $G$ : by definition, its unipotent radical $R_u(G)$ is trivial. One have ...
2
votes
2answers
97 views

Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
2
votes
1answer
69 views

Confusion about the quotient $G/B$

Let $G$ be an affine, complex, reductive algebraic group and $B$ a Borel of $G$. I have seen and understood the proof that $G/B$ is projective. Now, on the other hand, I have made the following ...
4
votes
1answer
77 views

Linearization of a group action: why the map is equivariant?

I'm using Dolgachev's book on invariant theory to learn linearizations of group actions. Here is a sketch of main construction: let linear algebraic group $G$ act on a quasi-projective variety $X$, ...
1
vote
1answer
51 views

Torus orbit closures and rank-1 subtori

Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$. Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in ...
2
votes
0answers
58 views

finite normal subgroup

$G$ is a subgroup of finite index in $SL(n,Z)$, $n\ge 3$, $N$ is finite normal subgroup of $G$, then I want to know why $N$ is a normal subgroup of $SL(n,Z)$. More generally, $A$ is an arithmetic ...
2
votes
0answers
41 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
4
votes
0answers
82 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
3
votes
0answers
133 views

Confused about Borel-Weil theorem

I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...
5
votes
2answers
137 views

What is the coordinate ring of $G/U$?

Let $G$ be an algebraic group and $U$ its subgroup consisting all upper triangular matrices. For example, $G=GL_n(k)$ and $U$ the subgroup consisting of all upper triangular unipotent matrices in ...
3
votes
1answer
35 views

Does the exponential map respect module actions?

Setup: Let $k$ be a field and $G \subseteq \mathrm{GL}_n(k)$ an algebraic group, reductive if that makes a difference. Let $\mathfrak g \subseteq \mathfrak{gl}_n(k)$ be the Lie algebra of $G$ with ...
1
vote
1answer
128 views

What is the group of $k$-rational points of an algebraic group?

Let $k$ be a field and $G$ a linear algebraic group over $k$. What is the group of $k$-rational points of $G$? By definition, $G$ is an algebraic variety. Suppose that $G$ is defined by polynomials ...
2
votes
1answer
92 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
0
votes
0answers
221 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
10
votes
0answers
150 views

a closed subset of an algebraic group with a constant tangent space is a coset

Let $G$ be an algebraic (not necessarily linear) group and let $Z \subset G$ be a Zariski closed irreducible subset. Since tangent bundle of $G$ is trivial, we may identify tangent spaces at all ...
1
vote
0answers
148 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
1
vote
0answers
77 views

P/B is isomorphic to the projective line $\mathbb{P}^1$

Suppose that $P \subset G$ is a parabolic subgroup containing a Borel subgroup $B$. Moreover, let $P$ be a minimal parabolic subgroup properly containing B, i.e., one corresponding to a single root ...
5
votes
2answers
236 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
2
votes
1answer
69 views

Factor/Quotient Group $M/\mathbb{Z}$

I've posted another question here a few days ago asking what a factor/quotient group is because I couldn't wrap my mind around it. Although I have an idea what it means, I still don't fully understand ...
3
votes
0answers
35 views

Closed subgroups of algebraic group have DCC?

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details? Prop: No. Proof: If ...
3
votes
1answer
165 views

Minimal parabolic subgroups of a reductive group - Bruhat type decomposition

Let $G$ be a reductive group, $B$ a Borel subgroup, $P$ a minimal parabolic subgroup having a Levi decomposition $P = UL$, let $\alpha$ be one of the two roots of $L$ relative to $T$, and $U_\alpha, ...
3
votes
1answer
66 views

Non-uniqueness of group structure for affine algebraic groups

We know that every abelian variety has a unique group structure, but in the affine case, is that every affine algebraic group has more than one (up to isomorphism) group structure?
1
vote
0answers
31 views

Baily Borel Compactification: choice of boundary

In Borel/Ji " compactifications of symmetric and locally symmetric spaces " the Baily Borel compactification of a locally symmetric space is defined as $$\Pi\backslash(X\coprod_{\bf{P}}X_{P,h})$$ ...
1
vote
0answers
55 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
1
vote
0answers
49 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...
0
votes
0answers
38 views

How does Weyl group acts on coroots?

We know that $W=N/T$ ($N = \{n \in G \mid nTn^{-1} = T \}$) acts on $T$ by $w(t)=wtw^{-1}$ (since $T$ is commutative, this action is well-defined). In the case of $GL_n$, by direct computation we know ...
4
votes
0answers
67 views

What is $\rho^{\vee}(-1)$?

What is $\rho^{\vee}(-1)$? By definition, $\rho^{\vee}$ is the sum of all positive coroots. I have some difficulty in computing $\rho^{\vee}(-1)$. For example, in the case of $SL_3$, all positive ...
1
vote
0answers
136 views

Possible small mistakes in Springer's *Linear Algebraic Groups*

In the 2nd edition of Springer's Linear Algebraic Groups, the proof of 16.2.2(i) (p. 271) begins with the assertion that the reductive group G is quasi-split over F iff the restriction map of ...
2
votes
0answers
126 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
7
votes
1answer
302 views

Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an ...
0
votes
1answer
43 views

Rigidity of Diagonalizable Algebraic Groups

This question is about a result in the section 16.3 of the book Linear Algebraic Groups from Humphreys. The follow can be deduced from a proposition in the section 16.3 of the book: Corollary: Let ...
3
votes
0answers
58 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
0
votes
1answer
51 views

Tate module of linear algebraic group

Let $G$ be a (smooth, connected, geometrically integral) commutative linear algebraic group over $\mathbf F_q$. Just as for abelian varieties, we can define the $\ell$-adic Tate module $$ T_\ell G ...
3
votes
1answer
306 views

How to compute radical and unipotent radical of an algebraic group?

By definition, an algebraic group $G$ is reductive if its unipotent radical is $\{e\}$. The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. The ...
1
vote
3answers
262 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
2
votes
0answers
26 views

Example of algebraic group of type $G_2$

Can anyone point me to a concrete realization of a reductive algebraic group of type $G_2$ over a field of positive characteristic? I have some questions about how the adjoint action permutes certain ...
1
vote
0answers
67 views

why is Borel subgroup not nilpotent?

Let $G$ be a simple linear group group over an algebraically closed field $k$, and let $B$ be a maximal solvable subgroup. If things are happening over $\mathbb{C}$ then I know how to show that $B$ ...
4
votes
0answers
111 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
3
votes
2answers
106 views

“$L/K$ forms of each other”

In section $4$ of these notes, the author says two algebraic groups $G$ and $H$ defined over a field $K$ are "$L/K$ forms of each other" if they are "isomorphic over $L$", where $L$ is a finite field ...
2
votes
1answer
59 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
3
votes
0answers
64 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
1
vote
0answers
55 views

Isomorphism types of stabilizers of vectors in linear representations of the special linear group

Suppose we have a linear representation of the group $SL_d$ over $\mathbb{C}$. i.e. a finite dimensional vector space $V$ with a linear action of $SL_d$ on it. Let $v\in V$ be some vector and let ...
3
votes
0answers
182 views

Exceptional isomorphisms of classical algebraic groups

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. An affine algebraic group $G$ is an affine variety over $k$ with a group structure such that multiplication and inversion are ...
2
votes
1answer
64 views

Integer subgroup of indefinite orthogonal group

I'm interested in the subgroup of the indefinite orthogonal group consisting of integer matrices i.e. $$ZO(p,q):=\{M \in GL(p+q, \mathbb{Z})| M^TI_{p,q}M=I_{p,q}\},$$ where $I_{p,q}$ denotes the ...
1
vote
0answers
34 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
4
votes
1answer
121 views

Is a finite normal subgroup of a reductive algebraic group central?

In a proof I am reading, the author considers the situation where $G$ is a reductive algebraic group (variety) over the complex numbers $\mathbb C$ and $N\trianglelefteq G$ is a closed, normal ...
2
votes
1answer
58 views

Finding the Hopf Algebra Coproduct coming from an Affine Group Scheme

I was wondering if anyone could help with how to, strictly from Yoneda's Lemma, obtain the coproduct map on the Hopf Algebra for an Affine Group Scheme. Particularly for something like $\text{SL}_2$ ...
1
vote
1answer
65 views

Automorphisms of algebraic groups

I am working through Humphreys' Linear algebraic groups, and I am stuck on the following exercise ( ex 4 on pg 57) I need to show that the only automorphisms of $G_m$ (as an algebraic group) is $x ...