All Questions
77
questions
3
votes
2
answers
172
views
The orbits of an algebraic action of a semidirect product of a unipotent group and a compact group are closed?
We consider real algebraic groups and real algebraic varieties. It is known that the orbits of an algebraic action of a unipotent algebraic group $U$ on an affine variety are closed. The orbits of an ...
- 199
1
vote
0
answers
116
views
Geometric induction of modules for algebraic groups
Let $\Bbbk$ be an algebraically closed field (of any characteristic). Let $G$ be an algebraic group over $\Bbbk$, and $H$ a closed (hence algebraic) subgroup of $G$.
Let $V$ be a finite-dimensional $...
- 1,077
2
votes
1
answer
397
views
Representation ring of the general linear group
The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of ...
- 611
3
votes
0
answers
110
views
Describing the outer automorphism of a special unitary group in terms of the Hermitian form
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
- 7,313
6
votes
0
answers
270
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,429
4
votes
1
answer
233
views
Bounded generation of group by unipotent radicals of opposite parabolic subgroups
Let $G$ be an almost $k$-simple group that is also simply connected (so that $G(k)^{+}=G(k)$). For opposite parabolic subgroups $P$ and $P^{-}$, it is known that $G(k)^{+}$ is generated by the ...
- 949
6
votes
0
answers
173
views
Functions of polynomial growth on linear algebraic groups
$\DeclareMathOperator\GL{GL}$Let $G$ be a complex linear algebraic group, i.e. a subgroup in $\GL_n({\mathbb C})$, defined by a system of polynomial equations
$$
p_i(x)=0
$$
(here $p_i$ are ...
- 7,366
4
votes
1
answer
239
views
Question regarding semistability of a point of GIT quotient
$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
- 585
2
votes
0
answers
125
views
Is the image of the exponential map of a complex semisimple group Zariski open?
Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map
$$\exp : \mathfrak{g} \to G$$
Zariski open in $G$?
- 443
2
votes
1
answer
228
views
An extension of algebraic torus
Let $T_1$ and $T_2$ be algebraic tori over a field of characteristic 0. Let $T$ be an extension of $T_1$ by $T_2$, namely
$$
1\longrightarrow T_1\longrightarrow T\longrightarrow T_2\longrightarrow 1.
$...
- 833
4
votes
1
answer
400
views
Universal covering groups of simple linear algebraic group schemes
Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e.,...
- 7,313
2
votes
1
answer
194
views
When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?
This is related to Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups. There I was told that in general, the symplectic group $\text{Sp}_{2n}(R)$ is not generated by its ...
user341790
3
votes
1
answer
260
views
Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups
This is a question about the answer in this other post: Symplectic group over integers and finite fields.
In general, for any ring $R$, the symplectic group $\text{Sp}(2n,R)$ is generated by its root ...
user341790
3
votes
0
answers
136
views
density of unipotent flows in algebraic groups
Let $\mathcal{G}$ be a reductive algebraic group over $\mathbb{Q}$ with a model $G$ over $\mathbb{Z}$ such that $G(\mathbb{R})$ is compact modulo centre. Let $T$ be a maximal torus of $\mathcal{G}$. ...
- 81
2
votes
1
answer
223
views
Properties of stabilizers of adjoint action general linear group
Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that ...
