Questions tagged [group-actions]
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479
questions
11
votes
3
answers
1k
views
Quotient of a smooth curve by a finite group and differentials
Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense ...
- 2,623
2
votes
0
answers
201
views
Manifolds as simultaneous coset spaces
Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...
- 151
3
votes
0
answers
241
views
Simultaneous coset spaces [closed]
Let $X$ and $Y$ be sets. Under what conditions is there a group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in $Y$?...
- 151
16
votes
1
answer
948
views
Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces
The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof.
Question: Let $M$ be a ...
- 4,868
3
votes
1
answer
163
views
How to construct a proper action of a group of finite virtual cohomological dimension?
Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$,
where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual ...
- 45
2
votes
1
answer
137
views
The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$
Motivated by the following RG question we ask a related question as follows:
We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes GL(\...
- 312
3
votes
2
answers
183
views
equivariant embeddings from the k-th configuration space to the k+1-th configuration space
Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
- 1,970
3
votes
1
answer
279
views
Averaging measurable functions over amenable group actions
Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
- 10.3k
3
votes
1
answer
234
views
On an example by Romagny about fixed point stack not commuting with coarse moduli space
This is to understand better Example 3.9 on page 221 of Group actions on stacks and applications by M.Romagny.
For an action of an algebraic group (scheme) $G$ on an algebraic stack $\mathcal{M}$, ...
- 22.7k
2
votes
2
answers
335
views
Invariant polynomials under the action of $H\le\operatorname{GL}_n(\mathbb{F}_p)$
Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}_n(\mathbb{F}_p)$ acts on the polynomial ring $\mathbb{F}_p[x_1,\ldots,x_n]$ via $A\cdot x_i=\sum_j a_{ji}x_j$ for ...
- 768
5
votes
0
answers
108
views
On finite quotients of unions of $n$ affine varieties
Assume that a finite group $G$ acts on a quasi-projective variety $Q$ (say, over complex numbers) that possesses a Zariski cover by $\le n$ affine varieties. My question is: does the quotient $Q/G$ ...
- 16.5k
9
votes
2
answers
2k
views
Which weighted projective spaces (and their finite quotients) are local complete intersections?
Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{...
- 16.5k
7
votes
2
answers
500
views
Algebraic proof without using comparison theorem for étale cohomology
Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity).
The étale cohomology groups of X are therefore equipped with an action of $\mu_n$...
- 2,501
4
votes
1
answer
168
views
Is this notion of 'closed subset' of a semigroup action something people have thought of?
Suppose $S$ is a semigroup (or a monoid, or a category), and $X$ is an $S$-set -- that is, a set with an action of $S$. Say that a sub-$S$-set $Y$ is "downward closed" (or maybe "well-generated") if ...
- 3,071
6
votes
2
answers
297
views
"Interesting" projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?
The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would ...
- 16.5k
3
votes
1
answer
133
views
symmetric group of regular polyhedrons
Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let $c:=c(...
- 543
14
votes
4
answers
1k
views
actions of the hyperoctahedral group
I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $...
- 5,563
2
votes
1
answer
310
views
Kunneth formula of Cartesian product modulo orders of coordinates
Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on
$$
\prod_n X
$$
by
$$
\sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in \...
- 2,213
0
votes
1
answer
312
views
Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?
$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times.
I am studying some function arising from symplectic geometry which happens in my case to be ...
- 1,873
4
votes
2
answers
814
views
Künneth formula for Bredon cohomology theory
Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...
- 745
7
votes
2
answers
866
views
Is there a topological Chevalley-Shephard-Todd Theorem?
Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( ...
- 635
3
votes
1
answer
190
views
Actions of the unit circle on finite complex matrices
Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle.
How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on $...
- 31
3
votes
1
answer
551
views
Equivariant Derived Category
Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.)
Suppose we have an algebraic group $G$ ...
- 335
3
votes
1
answer
708
views
Simultaneous action of GL(n) on matrices
Consider the action of $GL(n,k)$ on the set $M\times M$ where $M$ is the set of all $n$-by-$n$ matrices over $k$ given by $g\cdot(h,l) \mapsto (ghg^{-1}, glg^t)$.
Individually these actions are well-...
- 661
1
vote
0
answers
123
views
Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$
I ask this question in MSE and I received no answer, so I repeat it here:
Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...
- 312
4
votes
0
answers
239
views
Can Z/2 x Z/2 act freely on an infinite dimensional sphere?
Using that all groups that act freely on some sphere $S^n$ have periodic cohomology, one can see that $\mathbb Z/2 \times \mathbb Z/2$ can not act freely on any $S^n$. But can it act freely on $S^{\...
- 11.9k
6
votes
1
answer
404
views
Free actions of non-amenable groups
Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free (i....
- 1,210
6
votes
1
answer
219
views
Example of a $G$-sphere that is not a $G$-representation sphere
Let $G$ be a finite group with the discrete topology. To set terminology:
a $G$-sphere is a sphere equipped with a continuous $G$-action
a $G$-representation sphere is a $G$-sphere obtained from an ...
- 6,722
0
votes
0
answers
199
views
Action of semidirect products of cyclic groups
Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...
6
votes
2
answers
564
views
Stabilisers of group actions
Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.
Suppose there exists some point $x \in X$ whose stabiliser ...
- 20.8k
23
votes
4
answers
1k
views
Dividing by two in the category of vector spaces
Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$?
Here "naturally" means "in an $GL(V) \times GL(W)$-...
- 19.4k
3
votes
1
answer
708
views
finite stabilizers + compact orbit space => proper action?
Suppose a countable discrete group is acting on a smooth manifold with finite stabilizers and the orbit space is compact (and Hausdorff). How one can prove that the action is proper?
I found this ...
- 41
1
vote
0
answers
98
views
Name for condition on map of cancellative monoids
Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that
$k(\epsilon)=\epsilon$
for all $a,b\in M$, there exists $v\in N$ such that ...
1
vote
1
answer
248
views
The automorphism groups of smallest grammars of a language string are isomorphic
Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = \...
- 182
0
votes
2
answers
2k
views
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
3
votes
1
answer
1k
views
cohomology of the orbit space of a group action
Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...
- 1,970
5
votes
2
answers
367
views
fixpoint algebras of a permutation action
Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...
- 7,537
7
votes
0
answers
293
views
Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?
Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...
- 5,910
11
votes
1
answer
526
views
Is there a faithful transitive locally finite action of the modular group?
Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?
- 11.2k
2
votes
0
answers
212
views
Exotic actions of hyperbolic groups
Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that:
The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$.
For every quasiconvex subgroup $H \leq G$...
- 11.2k
9
votes
2
answers
367
views
Is there a highly transitive action of a finitely generated torsion simple group?
Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ?
Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two $k$-...
- 11.2k
2
votes
1
answer
1k
views
Coaction of a group
Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I ...
- 723
9
votes
2
answers
625
views
Action of the homotopy braid groups on reduced free groups
Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...
- 1,108
3
votes
1
answer
138
views
Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?
Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...
- 1,388
1
vote
0
answers
270
views
Adelic integral factorization
In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...
- 5,314
4
votes
1
answer
308
views
Free action of $\mathbb{Z}(2^{\infty})$ on a compact space
Assume that $X$ is a Hausdorff compact space such that $\forall n\in \mathbb{N}$, we have a free action of $\mathbb{Z/{2^{n}}\mathbb{Z}}$ on $X$. Must $\mathbb{Z}(2^{\infty})$ act freely on $X$?...
- 312
5
votes
0
answers
312
views
Unitary representations of Tarski Monsters and other beasts
Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
- 647
1
vote
0
answers
181
views
Invariant subsets of a local action
I have also asked this in MSE, but it seems to me that my question wasn't very well received there and I think someone in here will be able to answer it more quickly, hence this post.
I don't ...
- 170
2
votes
0
answers
304
views
Rational conjugation of elements of a finite group
Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...
- 21
13
votes
2
answers
616
views
Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$
I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
- 2,636