All Questions
2,543 questions
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Splitting of simply connected algebraic group
Let $k$ be a number field and let $G$ be a connected semisimple, simply connected algebraic group defined over $k$. Let $k'$ be a finite Galois extension over which $G$ splits. By the Chebotarev ...
- 223
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141
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Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
- 376
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121
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Commensurability of arithmetic, irreducible, nonuniform lattices
Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
- 255
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221
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Endomorphism rings of flat group schemes
Let $R$ be a commutative ring and $X$ be a flat $R$-group scheme.
We call $\text{End}_R(X)$ the ring of endomorphisms of the $R$-group scheme $X$, defined over $R$.
Let $R\to S$ be a ring map ...
user147445
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378
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Quotient of an affine scheme by an étale finite group
Let $G$ be a finite étale group scheme over a field $k$ and $X=\mathrm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\mathrm{Spec}(A^H)...
- 529
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79
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What is meant by "roots in $Lie(N)$" in root space decomposition of Lie algebras?
Let $G = GL_n$ and $T$ the invertible diagonal matrices and $N$ the upper triangular matrices with only $1$'s on the diagonal. Then the Lie algebra $\mathcal{G}$ has the roots space decomposition
$$
\...
- 3,625
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269
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How is this pairing $\langle\,,\rangle$ defined of cocharacter and character of an algebraic group?
Let $G$ be a semisimple linear algebraic group. Let $X^*$ be the group of characters and $X_*$ be the group of cocharacters. Then I know that there exists a pairing $\langle\,,\rangle : X^*(G) \times ...
- 3,625
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147
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Groups implementable by finite field
I'm interested in finding all groups for which the group operation (and inverse map) may be implemented using finite field arithmetic.
I've done some searching and have come across "algebraic groups",...
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103
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Is there a projection from $G$ to the Levi subgroup of a Parabolic subgroup?
Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel ...
- 6,211
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156
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Connected unipotent groups acting on an affine variety (re: stabilizers)
Let $U$ be a connected unipotent algebraic group over a field of characteristic $p>0$. Assume $U$ acts on an affine variety $X$ by regular maps.
Is it true that the stabilizers of rational points ...
- 993
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459
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Clarification on the definition of a smooth affine scheme over an integral domain
$\DeclareMathOperator{\Spec}{Spec}$
The following is from Bruhat and Tits article Groupes Reductifs sur un Corps Locale II. $A$ is an integral domain. Here $A$-scheme means "affine $A$-scheme," and $...
- 6,180
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126
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Books on algebraic groups over C with examples [duplicate]
I've been trying to learn about algebraic groups lately and I was wondering if there were any books/notes out there which: A. treat algebraic groups over the complex numbers, B. cover all the most ...
- 1,889
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256
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Bases of a relative root system are parameterized by what?
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus of $G$, and $\Phi = \Phi(T,G)$ the set of roots of $T$ in $G$. The bases $\Delta$ of $\Phi$ ...
- 6,180
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197
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'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
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283
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Normalizer of non-split tori
Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.
Question: What do we know about the normalizer $N_G(T)$...
- 2,751
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answer
139
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presentation for a nilpotent group associated to the square of a coxeter element
This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups.
Let $\Gamma$ be a coxeter diagram --- i.e. an unoriented graph with $r$ vertices ...
- 2,137
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138
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Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series
Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
- 1,702
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121
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Non-split simple groups
Let $G$ be a simple group over a field $F$, and let $K$ be an extension of $F$ such that $G$ base changed to $K$ is split. Does this mean that $K$ splits a maximal torus of $G$?
Context: I want to ...
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103
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Relation between $\Gamma$-percuspidal parabolic subgroups and split parabolic subgroups of real semisimple Lie groups
Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. Let $\Gamma$ be a lattice in $\mathcal{G}:= G(\mathbb{R})$. I am interested in knowing under what conditions on either of $\mathcal{G}...
- 61
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161
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Rational group scheme
Suppose $G$ is a group scheme over a field $k$, i.e., $G$ is a functor from the category $\text{Alg}_k$ of unital commutative, associative $k$-algebras to the category of $\text{Groups}$. Suppose that ...
user69002
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66
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Invariant subalgebra and dual torus for symmetric group
Given permutation module with three generators and corresponding Galois action of symmetric group $\mathfrak S_3$ I am interested in computing corresponding dual torus $T$ (which should be of ...
- 23
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259
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Zariski dense subgroups and conjugates
Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
- 11.3k
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400
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Can any reductive $k$-group be written as a semidirect product of $k$-linear groups?
This might be an easy question for the experts, so I apologise in advance. By a reductive group over a field $k$, I mean a linear algebraic group (not necessarily connected) such that the unipotent ...
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153
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torsors on quasi-split groups
Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$.
Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$.
We consider $G$ a quasi-split group scheme over $X$ ...
- 3,472
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162
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What algebras does the hidden subgroup problem for finite abelian groups apply to?
Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...
- 223
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144
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The centralizer $Z_G(X)$ of a nilpotent element in a real simple Lie group
I am looking for the description of the centralizer $Z_G(X)$ , where $G$ is a real simple Lie Group and $X\in \ Lie (G) $ such that $X^d=0,\ X^{d-1}\neq 0 $. It is is helpful to me any references or ...
- 116
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257
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rational conjugacy classes
Say that $G$ is a reductive group over a local field $F$ and $g\in G(F)$. Can we show that the conjugacy class of $g$ in $G(F)$ has finite index in the $F$-rational part of the conjugacy class of $g$ ...
- 2,125
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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
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333
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on the Galois cohomology of reductive groups
Let $G$ a simply connected group over an algebraically closed field.
$F=k((t))$ and $\mathcal{O}=k[[t]]$.
Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$.
Let $E=k((t^{1/n}))$ with $n$ prime to the ...
- 3,472
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115
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a very elementary question on the conjugated matrices
Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero .
We suppose that they have the same ...
- 3,472
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109
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solve the singularities of parabolic orbits of schubert cells
Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}.
For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of $X_{w}=\...
- 3,472
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0
answers
273
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
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0
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440
views
Foliations in positive characteristic
Ekedahl wrote about foliations in positive characteristic, over the field $\mathbb{Z}/p\mathbb{Z}$ as a subsheaf of the tangent sheaf, that are closed with respect to involution and $p$-power.
My ...
- 1
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0
answers
129
views
A kind of orthogonal subgroup
Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product.
Is it true that $\{x \in \...
- 285
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0
answers
2k
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Book on linear algebraic groups in scheme language
Is there a book on linear algebraic groups using the scheme language (i.e. not Springer or Borel, but like Waterhouse, but more in-depth)?
The book should discuss topics like Borel subgroups etc.
(...
- 417
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votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
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524
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DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
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839
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intersection of a parabolic subgroup with a subgroup
I'm interested in the following question: let $k$ be a field of characteristic zero (just for simplicity), $G$ a connected semi-simple $k$-group, $P\subsetneq G$ a parabolic $k$-subgroup, and $H\...
- 313
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700
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Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
- 2,709
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1k
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The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)
Original question (without additional information from Wendy):
Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:
Taking the E8 as {128,...
- 203
-1
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1
answer
2k
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Semisimple elements of a lie algebra
Let $G\subset GL_n(\mathbb{C})$ be an algebraic group of dimension n, and let $\mathfrak{g}$ its Lie algebra.Is there a relations between the maximal number of independent semisimple elements of $G$ ...
- 671
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1
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236
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$A[x]$ points of an algebraic group
Let $K$ be a field and $G$ be an algebraic group. Specifically $O(n)$ or $Sp_{2n}$. Is it true that for any ring $A$ over $K$ , $G(A)\cong G(A[x])$.
Is there any reference for such kind of results?
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$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes
Edited 1/21/2018 to add the following:
Here is a DropBox link
https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0
to a PDF showing how my team used biomolecular first ...
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