# Tagged Questions

**5**

votes

**1**answer

216 views

### Index of congruence modular subgroup of level (1,d)

Let $D = \text{diag}(1,d)\in M_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma_D\subset M_{4}(\mathbb{Z})$ as:
$$\Gamma_D := \left\lbrace R\in ...

**3**

votes

**1**answer

357 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

**3**

votes

**2**answers

169 views

### Universal Central Extension of pi(X), X a compact Riemann surface of genus>1

Does a universal central extension exist for the fundamental group of a Compact Riemann Surface of genus1? Please give a detailed explanation.I am unable to justify the statements in Atiyah-Bott Phil ...

**3**

votes

**3**answers

438 views

### A neat monodromy group of a family of Kaehler manifolds

Let $X\rightarrow B$ be a family of Kaehler manifolds with possibly singular fibers. Let $G$ be the monodromy group on $H^n(X_b,\mathbb{Z})$, where $n=\dim X_b$ with the smooth fiber $X_b$ over some ...

**0**

votes

**0**answers

48 views

### Finding stable ideals of F_3[[X,S]] by group action.

Let k > 1 be a positive integer and define the action σ_k on F_3[[X,S]] by
σ_k: X ---> X + S + X^k
σ_k: S ---> S + S^3.
Then,
Conjecture: There exists a principal ideal (a) other than (S) such ...

**0**

votes

**0**answers

92 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 ...

**3**

votes

**3**answers

326 views

### Reductive groups over finite fields

Let $\ell\ge 5$ be a prime number. Let $G/\mathbb{F}_\ell$ be a (smooth, connected) reductive algebraic group. Let $G(\mathbb{F}_\ell)^+$ be the normal subgroup of $G(\mathbb{F}_\ell)$ generated by ...

**2**

votes

**1**answer

359 views

### A question about flag variety of $SL(n,\mathbb{C})$

We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of ...

**7**

votes

**2**answers

600 views

### Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?

Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , ...

**2**

votes

**1**answer

280 views

### Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...

**0**

votes

**0**answers

55 views

### Geometric effects of removing elements of D2n generalizable?

So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...

**5**

votes

**1**answer

391 views

### Deformations of p-divisible groups

Given a p-divisible group over $\mathbb{F}_p$, Grothendieck-Messing theory tells us that deforming the group to $\mathbb{Z}_p$ is the same as finding an admissible filtration of the Dieudonne-module ...

**2**

votes

**2**answers

380 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 ...

**4**

votes

**0**answers

265 views

### Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...

**2**

votes

**0**answers

78 views

### Bisections in Kan Complexes

Kan Complexes can be seen as a generalization of groupoids, mostly called (weak)
infinity groupoids in this context.
On groupoids we can define the \textbf{group of bisections} the following way:
...

**10**

votes

**2**answers

422 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 ...

**1**

vote

**1**answer

198 views

### Fixed points of group action

Let us consider the group $PGL(2,\mathbb{R})$ as the group of automorphisms of real projective line and $H\subset PGL(2,\mathbb{R})$ is a subgroup of prime order $> 2$. Is it true that there always ...

**15**

votes

**2**answers

1k views

### What is a Gaussian measure?

Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...

**4**

votes

**3**answers

287 views

### Spectrum and scheme of the commutative group-algebra of an abelian group.

The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...

**0**

votes

**0**answers

72 views

### on trivialisation of T-torsors

Let $X$ a smooth connected projective curve over an algebraically closed field $k$ and $F$ its function field. $T$ a $X$-torus.
Let $R$ be any ring.
Let $E$ a $T$-torsor on $(X-x)\times_{k}R$. Does ...

**8**

votes

**2**answers

351 views

### Birational Automorphisms and infinite divisibility

Suppose $X$ is some algebraic variety. It can be over $\mathbb{C}$, but it doesn't have to (but char $0$ preferred).
Is it possible that the additive group $\mathbb{Q}$ acts on it birationally, ...

**4**

votes

**3**answers

257 views

### Automorphism of finite groups and Hurwitz spaces

If $G$ is a finite group, embedded as a transitive subgroup of $S_n$ for some $n$, will every automorphism of $G$ extend to an inner automorphism of $S_n$?
I'm trying to connect the language that's ...

**2**

votes

**3**answers

374 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 ...

**12**

votes

**0**answers

584 views

### The symmetric group and the field with one element

I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of ...

**0**

votes

**1**answer

141 views

### subgroups of a $p$-solvable group and complete reducibility

1.
Let $G$ be a $p$-solvable group and $V$ be a finite dimensional
faithful $kG$-module, where the characteristic of $k$ is $p$. But
$V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...

**0**

votes

**0**answers

95 views

### 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 ...

**2**

votes

**1**answer

147 views

### resolution of strata of the affine grassmanian

Let G a semisimple simply connected group over an algebraically closed field.
Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter
...

**6**

votes

**1**answer

581 views

### Groups becoming algebraic groups

Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that:
(1) the set of $k$-points has the structure of a group.
(2) for any $g\in G$ the ...

**5**

votes

**1**answer

371 views

### Confusing Point in Proof: Semisimple Automorphism Fixes Torus

I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof.
The setting is as follows. ...

**3**

votes

**0**answers

300 views

### Does this property of subgroups (or sheaves of ideals) already have a name?

I'm constructing an example of a group which has a particular property on its subgroups, and the property looks like something that might have been considered before.
Fix a group $G$ and a pair of ...

**2**

votes

**2**answers

419 views

### elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...

**1**

vote

**1**answer

150 views

### fixed point scheme in caracteristic p

Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k.
Let G a finite group acting on X and suppose that his order is divisible by the caracteristic ...

**4**

votes

**0**answers

120 views

### Is there an arithmetic analogue of Drinfeld's count of a number of 2d irreps of fundamental group of a curve ?

There is a paper by V. Drinfeld 1981, which title is "Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field".
It gives a formula for this ...

**2**

votes

**1**answer

248 views

### smooth group scheme genrically abelien

Let J a smooth group scheme over a smooth connected base S.
I assume, that over an open subset U of S, J is a torus, do I have that J is abelian?

**13**

votes

**1**answer

549 views

### Motivation behind the construction of Deligne and Lusztig

If $G$ is a connected reductive group over a finite field $\mathbb{F}_q$ and $T$ is a maximal torus in $G$, the famous construction of Deligne and Lusztig (Annals of Math, 1976) associates ...

**1**

vote

**0**answers

83 views

### Equivalence between Algebraic Semi-group Structures and Coalgebra Structures for an Algebraic Variety?

I was looking at this old question
Hopf algebra and group structure correspondence for algebraic varieties
which says that there exists an equivalence between algebraic group structures on
an ...

**3**

votes

**2**answers

298 views

### Configuration of the branch locus of a branched covering of an elliptic curve

Let $C$ be a curve of genus 3 and suppose that it admits a branched cover $\varphi:C\rightarrow E$ with $E$ elliptic and such that $\varphi$ does not factor through any \' etale cover. Then the degree ...

**18**

votes

**2**answers

1k views

### Criteria for irreducibility of polynomial

If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best

**0**

votes

**1**answer

284 views

### For an algebraic group acting on a variety, why are orbits representable?

I suspect this is really obvious, but I'm not seeing it.
For an algebraic group $G$ acting on a variety $V$, and for a point $x \in \text{hom}(\text{Spec}(K),V)$, we define the orbit $G(x)$ to be ...

**3**

votes

**1**answer

256 views

### When does dimension behave nicely for quotients of affine algebraic varieties by the action of a group?

Let $X$ be an affine algebraic variety over an algebraically closed field $k$ of characteristic zero. Let $G$ be a reductive algebraic group acting on $X$. In this setting, there exists a categorical ...

**8**

votes

**1**answer

340 views

### Character varieties of finitely generated groups

Consider the following situation: $\Gamma_0\leq\Gamma$ are both finitely generated groups and $\Gamma_0$ has finite index in $\Gamma$. The restriction gives a well defined map between the character ...

**2**

votes

**2**answers

400 views

### The group G^+ of algebraic groups over local fields

Let $G$ be an algebraic group defined over a char 0
local field $k$. Following Borel and Tits (73) we define
the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$
generated by the unipotent elements ...

**0**

votes

**1**answer

194 views

### eigen-bundles of a trivial vector bundle

Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on ...

**6**

votes

**2**answers

356 views

### Generalized Grassmannians that parameterize the submodules of a module

I'm looking for something like a Grassmannian, but which parameterizes the submodules of a module rather than the subspaces of a vector space. Most specifically, I'm looking for something which ...

**5**

votes

**1**answer

417 views

### Hurwitz's automorphisms theorem with deformations

Hurwitz's automorphisms theorem bounds the order of the automorphism group of a negatively curved Riemann in terms of the genus.
Now suppose a finite group $G$ acts faithfully on a Riemann surface ...

**3**

votes

**1**answer

215 views

### Is the image of the representation of the fundamental group associated to a local system discrete?

If $f: X \to S$ is a projective smooth morphism between complex algebraic varieties. Does the $\pi_1(S)$-representation corresponding to the local system $R^i f_* (C_X)$ on $S$ maps $\pi_1(S)$ onto a ...

**43**

votes

**2**answers

2k views

### a categorical Nakayama lemma?

There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...

**5**

votes

**1**answer

186 views

### Finite index subgroups of the mapping class group with geometric meaning

I have got a question that is perhaps not precise in a mathematical sense.
Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...

**4**

votes

**1**answer

486 views

### Random geometries

Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all ...

**13**

votes

**2**answers

474 views

### A space of ideals

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...