Questions tagged [chevalley-groups]
The Chevalley group is a way, uniform over all fields (and commutative rings), to define a split simple algebraic group of a given type.
32
questions
8
votes
1
answer
782
views
There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$
Let $G$ be a split reductive algebraic group (over a local field if you like), $B$ be a fixed Borel subgroup, and $P$ be a fixed standard parabolic subgroup. Let $W$ be the Weyl group of $G$. For $w\...
- 940
7
votes
1
answer
402
views
Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups
I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
- 332
7
votes
0
answers
2k
views
What meanings does "Chevalley group" have?
It appears to me that there are at least two working definitions of the term "Chevalley group" operative in the literature. For example, one can consider Steinberg's notes on the subject. Starting ...
- 590
6
votes
1
answer
284
views
Number of points on a linear algebraic group over a finite field
Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$?
One can get something fairly nice ...
- 19.1k
6
votes
0
answers
146
views
Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?
The hyperbolic orthogonal group $O_{g,g}(\mathbb{Z})$ often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let $H$ denote the lattice $\...
- 7,843
5
votes
1
answer
214
views
Signs in Chevalley systems for reductive groups
Let $G$ be a pinned split reductive group. There exists a Chevalley system:
For each root $b$ in its root system there are parametrisations $x_b: \mathbb{G}_a \rightarrow U_b$ of the corresponding ...
- 51
5
votes
0
answers
164
views
Homology of a semisimplicial scheme
This is a question about the homology of a complex made of algebraic varieties. Consider the following subgroups of $\mathrm{SL}_3$ (defined over $\mathbb{Z}$).
$$
P_{1,2} = \left\{\left(\begin{...
- 2,030
4
votes
1
answer
196
views
Chevalley groups $G_{\mathbb{F}_2}$ in $G_\mathbb{Q}$
Is every (adjoint) Chevalley group over the field with two elements $G_{\mathbb{F}_2}$ isomorphic to a subgroup of its counterpart over the rationals $G_\mathbb{Q}$?
- 2,613
4
votes
1
answer
180
views
Schur multiplier of a Chevalley group of type $D_5$
$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $...
- 397
4
votes
2
answers
135
views
Twisted root subgroups in twisted Chevalley groups (reference request)
I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.
Let me first recall the classical set-up. According to Steinberg'...
- 167
4
votes
0
answers
295
views
Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring
Given a semisimple Lie algebra $\frak{g}$ of type $\Phi$ with a Lie algebra representation $\rho:\frak{g}\to \frak{gl}(v)$ and an arbitrary commutative ring one can associate the following gadgets:
...
- 405
4
votes
0
answers
223
views
Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?
Let $G$ be a complex Chevalley group (not necessarily adjoint type) with $\operatorname{\mathbb{C}-rank}\geq2$ and let $H$ be a normal subgroup of $G(\mathbb Z)$ with a finite index (so $H$ is Zariski ...
- 71
4
votes
0
answers
134
views
Exotic 2-adic lifts of mod $2$ Steinberg idempotent
Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...
- 3,031
3
votes
1
answer
264
views
For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
- 332
3
votes
1
answer
372
views
Conjugation of root subgroups by the Weyl group
Fix a field $k$ of characteristic zero, and let $G$ be a connected reductive algebraic $k$-group of isotropic rank $\ge 1$. Fix a maximal $k$-split torus $S$, and let $\Phi_k$ be the relative root ...
- 215
3
votes
0
answers
281
views
Representations of GL(n,2) over a field of characteristic 2
I would appreciate very much if you can point to me some references on the following:
1) Representations of the linear group $GL(n,2)$ over $F_2$.
2) Representations of $GL(n,2)$ over an algebraic ...
- 377
2
votes
1
answer
161
views
If a Weyl element preserves a root, then it has a representative which preserves the root space?
Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...
- 940
2
votes
1
answer
372
views
Maximal torus of Chevalley group $Sp(4)$
Consider a chevalley group a field $K$, with the right chevalley basis. Let $\alpha$ be a root. Let $x_{\alpha}(t)$ be the corresponding root space. Define $w_{\alpha}(t)=x_{\alpha}(t)x_{-\alpha}(-t^...
- 685
2
votes
1
answer
372
views
Diagonal automorphisms for twisted Chevalley groups
Let $G$ be a Chevalley group over a field $k$ of characteristic $0$. We know that a diagonal automorphism $\phi_h$ of $G$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ ...
- 41
2
votes
1
answer
201
views
How to prove that Chevalley groups over $\mathbb R$ have no compact factors
I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem.
I've been told in another thread ...
- 332
2
votes
1
answer
250
views
Root subgroups of simply connected Chevalley groups and their generators
I am looking for a detailed mapping of the root subgroups and elements (and their height) of the simply connected Chevalley groups of type other than $A_n$, and their generators into $\operatorname{GL}...
- 332
2
votes
1
answer
226
views
Character of a semisimple connected Lie groups [closed]
I'm trying to see why the Chevalley groups over $\mathbb C$ have no nontrivial character?
I know that a compact connected semisimple Lie group has no nontrivial character but is the compactness ...
- 332
2
votes
0
answers
105
views
A group generated by all the root subgroups above ideal of $\mathbb Z[\alpha]$ is of finite index in $G(\mathbb Z[\alpha])$
Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.
From "A note on generators for arithmetic subgroups of algebraic groups" by ...
- 332
2
votes
0
answers
168
views
Characterize an element of $\operatorname{SL}_n(\mathbb Z)$
I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$.
In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where
$$e_{1,n}(m)=
...
- 332
1
vote
1
answer
142
views
Maximal split torus of universal chevalley group
Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by $\{h_{\...
- 685
1
vote
1
answer
222
views
Universal Chevalley group associated to $D_l$
Consider the simple Lie algebra $D_l$. Consider the universal Chevalley group $G$ over a field $K$ associated to it. Then $G$ is a subgroup of the orthogonal group $O_{2l}(K, f)$ where $f$ is the ...
- 11
1
vote
1
answer
187
views
If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$?
If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is ...
- 332
1
vote
0
answers
104
views
Reference request: Commutator relations for the exceptional group F4
Is there any standard reference for the commutator relations for the exceptional group of type $F_4$?
If this question is not appropriate here, please let me know and I will delete it.
Thanks in ...
- 940
1
vote
0
answers
57
views
Representations of Chevalley algebras over arbitrary fields
In professor Humphrey’s “Introduction to Lie Algebras and Representation theory” it is explained how we can reduce a semisimple complex Lie algebra (and its representations) to an arbitrary field.
...
1
vote
0
answers
235
views
Bruhat cell of a Coxeter element
If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a ...
- 332
1
vote
0
answers
316
views
Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$
Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
- 332
1
vote
0
answers
268
views
(When) can the presentation in Steinberg's Yale notes fail to give an algebraic group?
I'm trying to understand a remark which appears on p. 1483 of Cohen, Murray and Taylor's "Computing in Groups of Lie Type." It says, "We have not used the presentations described in [7] or [30] ...
- 590