All Questions
74 questions
7
votes
2
answers
231
views
Non-semisimple symmetric subgroups of simply connected simple algebraic groups
Let $G$ be a simply connected simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a symmetric subgroup of $G$. This means that there exists an automorphism of order 2 $\...
- 14.2k
3
votes
2
answers
555
views
Counterexamples to Margulis Normal subgroup theorem in rank 1
Margulis' normal subgroup theorem states that any normal subgroup of a higher rank irreducible lattice is either finite or of finite index.
What are the known counter-examples in rank $1$ ?
I am ...
- 25k
6
votes
1
answer
390
views
Can these two irreducible $GL_n \mathbb Z$-representations be isomorphic?
Fix $n\in \mathbb N$ and a partition $\lambda$ with at most $n-1$ parts (of length at most $n-1$). Let $V$ be the irreducible $GL_n \mathbb R$-representation with highest weight $\lambda$ and $D$ the ...
- 156k
2
votes
4
answers
980
views
About structure of parabolic subgroups of finite classical algebraic groups
I am interested about a Fact (if it is right) of the structure of parabolic subgroups of finite classical algebraic groups:
Let $G$ be a classical algebraic group over the finite field of order $r^{e}...
- 26.3k
7
votes
1
answer
442
views
Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?
Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?
- 3,911
5
votes
2
answers
505
views
A finiteness property for semi-simple algebraic groups
Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any semi-...
- 8,529
20
votes
2
answers
1k
views
Solving equations in SO(3) : an open problem by Jan Mycielski
I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, http://www.jstor.org/stable/...
- 25.5k
6
votes
1
answer
214
views
Are the integer matrices in SO(3,2) "boundedly generated"?
Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$.
(The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...
- 3,911
1
vote
1
answer
167
views
Finite groups normalizing a torus
Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...
- 149k
10
votes
1
answer
719
views
what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?
Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
- 68.9k
4
votes
1
answer
1k
views
Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,612
0
votes
1
answer
208
views
on lifting extensions
Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...
- 1,285
24
votes
3
answers
2k
views
Spin group as an automorphism group
Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with $p$...
- 4,008
0
votes
0
answers
168
views
semisimple conjugacy classes over general bases
Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if $\gamma,\gamma'\...
- 3,472
1
vote
0
answers
508
views
on the open bruhat cell
Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de Concini-...
- 3,472
2
votes
0
answers
212
views
Compute the discriminant for reductive groups
Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
- 3,472
3
votes
2
answers
2k
views
Is there an almost-direct product decomposition for disconnected reductive algebraic groups?
$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
CommunityBot
- 1
11
votes
2
answers
1k
views
Finite subgroups of $PGL(3,K)$
It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
CommunityBot
- 1
1
vote
1
answer
376
views
on z-extensions
Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a $...
- 14.2k
4
votes
2
answers
1k
views
Dimension of Unipotent Radicals
A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
CommunityBot
- 1
0
votes
0
answers
272
views
minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,472
2
votes
2
answers
757
views
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 $\mathrm{Aut}(\mathbb{Z}^n)\...
- 8,493
4
votes
2
answers
578
views
Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 108k
10
votes
2
answers
1k
views
Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"?
At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
- 316