All Questions
2,543 questions
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Compactifications of group schemes
Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
- 41
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307
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For what fields is $GL_n(k)$ a rational variety?
I know that every linear algebraic group is rational over algebraically closed fields. To what extent is that true for other fields? For example: is $GL_n(\mathbb{Q}_p)$ a rational variety? Are there ...
- 11
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88
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open immersion, affine grassmanian and negative loop group
Let $G$ a semisimple group over $k=\bar{k}$.
Let the $k$-indgroup, $LG^{-}\subset G(k[t^{-1}])$ be the kernel of the reduction. We know by Faltings that the multiplication map:
$LG^{-}\times G(k[[t]]...
- 3,472
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102
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do commutative groups torsors have a point in an Abelian extension of the base field?
Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...
- 4,070
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120
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Base change of affine group schemes with respect to Frobenius map.
Suppose $G$ is an affine group scheme over a perfect field $k$ of characteristic $p>0$. Let $G^{(p)}$ be the base change of $G$ with respect to the Frobenius map of $k$ (i.e. $p$-th power map). Is ...
- 97
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132
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On the explicit formula of the height function occuring on the doubled Weil representation.
Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this.
Let $F$ be the ...
- 63
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97
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Degree of a commutator in a hyperalgebra or enveloping algebra
Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...
- 3,637
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301
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How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165
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70
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reduced group covers of a curve
Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a ...
- 314
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203
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Frobenius kernel for unipotent algebraic groups
Let $G$ be a connected algebraic group in positive characteristic $p$. If the Frobenius kernel $G_{(p)}=ker (F:G\to G^{(p)})$ is unipotent, do we have $G$ also unipotent?
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220
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Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
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82
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decomposition lemma in adelic groups II
Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.
Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.
On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}...
- 3,472
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140
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on a decomposition lemma in adelic groups
Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...
- 3,472
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192
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"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
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269
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how to determine the Weyl group of a diagonalizable subgroup?
Assume that $G$ is a connected compact Lie group (or a connected complex reductive group), and $K$ is a diagonalizable subgroup of $G$. It is known that the Weyl group $W_G(K)$ of $K$, defined as $N_G(...
- 11
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238
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Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
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617
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levi subgroup generated by maximal tori?
In the levi decomposition of an connected algebraic group $G$, is the levi subgroup generated by maximal tori of $G$?.
- 11
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116
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Reference request: a verification of a nonstandard subgroup being a Tits subgroup.
I have a particular infinite-index subgroup $H$ of the genus 2 symplectic group $Sp(2, \mathbb{R})$. This subgroup is self-normalizing (ie. $gHg^{-1}=H$ only if $g\in H$). I am looking to determine ...
- 2,274
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457
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Why do twists of an algebraic group over k correspond to k-torsors over G
Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$.
I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$.
It'...
- 1,213
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189
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Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
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157
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On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
- 11
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300
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When does a group action on $X$ preserve the reduction $X_\text{red}$? [duplicate]
Possible Duplicate:
Does the action of an affine group scheme preserve the nilradical of an algebra?
Let the group scheme $G$ act on the scheme $X$. I labored for a time under the misapprehension ...
- 27.9k
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313
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two different properties for the quotient
(Updated)
I have looked the draft of Ch4 of the book "Abelian Varieties" by Gerard van der Geer and Ben Moonen. It looks like in order to see the group scheme structure on G/H, one should consider ...
- 1,109
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409
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Pushforward of equivariant bundles via the Frobenius morphism
Let $G$ be a semisimple algebraic group over an algebraically closed field of positive characteristic $p$ and let $B \subseteq G$ be a Borel subgroup. Set $X := G/B$, the flag variety of $G$. Also let ...
- 3,637
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229
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twisted forms of a given group embedded in a second group?
Consider the following question about forms of a given group that are embedded in a fixed group.
Fix for simplicity $k$ a perfect field, and $H\subsetneq G$ a pair of connected reductive $k$-groups, ...
- 1,305
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418
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Centralizers and Cartan involutions
This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.
Consider $G$ a connected non-compact semi-simple Lie group, ...
- 313
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186
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Duflot-type theorem for Hopf algebras ?
In group cohomology Duflot's theorem states that the depth of the mod p cohomology ring of a finite group is greater than or equal to the p-rank of the center of a Sylow p-subgroup.
Is there a ...
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3
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678
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How do I show that a separable isogeny is central?
I've been trying to prove this (probably very simple) result that is stated in a paper that I'm reading:
Let $G$ and $H$ be connected semisimple algebraic groups defined over a field $F$, and let $f: ...
- 47
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2
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2k
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Connected components of algebraic groups
Let $G$ be an algebraic group, and $G_{Id}$ the connected component of the identity. Then $G_{Id}$ is a normal subgroup of $G$ and $G/G_{Id}$ is the component group of $G$.
Let $G_{c}\subset G$ be ...
user58018
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1
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470
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Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
- 1,783
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2
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428
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Group actions on affine space which are almost good
Let $G$ be a finite group acting on $\mathbb A^n_{\mathbb C}$. Let $Y$ be a dense open whose complement is of codimension at least two.
Assume $Y$ is $G$-stable, the action of $G$ is free on $Y$, ...
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678
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For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?
Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi ...
- 35
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2
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298
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quasi-affine-ness [closed]
Let $G$ be a group. Let $H$ be a subgroup of $R_u(G)$. Then $G/H\rightarrow G/R_u(G)$ is a $R_u(G)/H$ fibration. It is well known that
$R_u(G)/H=\mathbb{A}^n$. Is $G/H$ a quasi-affine variety?
user111251
0
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1
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183
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Slicker computation of the Lie algebra of the symplectic group (and computing differentials of matrix equations of polynomials)
Let $\mathbb{k}$ be an algebraically closed field. The symplectic algebraic group is given by
$$
\text{Sp}(2n,\mathbb{k})=\{M\in\text{Mat}_{2n}(\mathbb{k})\mid J=M^TJ M\}\quad\text{where}\quad J=\...
- 980
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1
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641
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Torsors and Central Extensions
In the setting of algebraic groups:
I understand that a central extension of a group $G$ by an abelian group $A$ is a exact sequence of groups :$0\rightarrow A\rightarrow \tilde{G}\rightarrow G\...
- 335
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1
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374
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Existence of a model over S-integers
Let $G$ is a connected reductive group over a number field $F$. Let $S$ be a set of places containing the archimedean places for which $G$ has a model over the ring of $S$-integers in $F$. Is it true ...
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1
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124
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Horospherical type of a spherical variety
In the following, I will fix $k$ a characteristic zero algebraically closed field, and $G$ a connected reductive group over $k$, $B$ a Borel subgroup of $G$, $T\subseteq B$ a maximal torus, $X$ a $G$-...
- 121
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146
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Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
- 479
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1
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187
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Algebraic Groups of Type H_3 and H_4 [closed]
By coincidence i stumbled over this page
http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/abstracts.html
, which was installed for a workshop on algebraic groups in 2012.
In the ...
- 1,479
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1
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245
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compact subset in linear algebraic group over local field
Let $F$ be a local field of characteristic zero, and $G$ a linear algebraic group with finite connected components over $F$. We will consider the $G(F)$, and give it $p$-adic topology. Let $C$ be some ...
- 2,709
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votes
2
answers
386
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Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 75
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1
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203
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Action of $SL_{n+1}$ on couples of linear spaces in $\mathbb{P}^n$.
Let $SL_{n+1}$ act on $\mathbb{P}^n$ in the natural way. Suppose I take two linear subspaces $\mathbb{P}^m$ and $\mathbb{P}^{n-m}$, with $m < n$, that intersect in one point. Is the action of $SL_{...
- 3,779
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166
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Calculating relative root systems
Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. ...
- 43
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1
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573
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When is the set of $n$-th powers in a group a subgroup? [closed]
Let $G$ be a non abelian group and $G_n=\{x^n | x\in G\}$ and n is integer. Is there a sufficient condition that makes $G_n$ be a subgroup of $G$ for arbitrary $n$?
- 101
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105
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About the connection between repellents and attractors under a $\mathbb{C}^{*}$ action on a projective variety
Let $X$ be a smooth projective variety with an action of $\mathbb{C}^{*}$. Let us suppose that the set $X^{\mathbb{C}^{*}}$ is finite. For $x \in X^{\mathbb{C}^{*}}$, let $A_{x}$ denote the attractor (...
- 103
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1
answer
185
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Strongly Irreducible Action implies Semisimplicity
Let $G$ be an algebraic closed subgroup of $SL(n,\mathbb{R})$ whose action on $\mathbb{R}^n$ is strongly irreducible, i.e. there is no finite union of proper nonzero linear subspaces of $\mathbb{R^n}$ ...
- 615
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1
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252
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Are Zariski connected and closed semisimple subgroups of semisimple and simply connected algebraic groups again simply connected? [closed]
Let $G$ be a semisimple and simply connected linear algebraic group over $\mathbb{C}$.
Let $H$ be a connected, Zariski closed and semisimple linear algebraic $\mathbb{Q}$-subgroup of $G$.
Is $H$ a ...
- 111
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1
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270
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A bijection between Lusztig series induced by inflation
Context:
Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F \...
- 309
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1
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208
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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 ...
- 3,472
0
votes
2
answers
212
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A kind of orthogonal subtorus
Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup
$S = \{x \in \...
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