Questions tagged [group-actions]
The group-actions tag has no usage guidance.
467
questions
67
votes
3
answers
20k
views
Properly Discontinuous Action
When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
- 1,216
65
votes
9
answers
9k
views
List of Classifying Spaces and Covers
I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a ...
31
votes
7
answers
5k
views
Invariant polynomials under a group action (hidden GIT)
Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}_n$ ...
- 1,953
29
votes
4
answers
7k
views
Intuitive explanation of Burnside's Lemma
Burnside's Lemma states that, given a set $X$ acted on by a group $G$,
$$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$
where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of ...
- 6,702
24
votes
1
answer
684
views
Given a group action on a simplex, can I always find a fundamental region that is a simplex?
Let $\Delta\subset\Bbb R^n$ be a simplex with $n+1$ vertices. Let $G\subset\mathrm{GL}(\Bbb R^n)$ be a finite group of linear symmetries of $\Delta$, i.e. linear transformations that fix the simplex ...
- 11.9k
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 $GL(V) \times GL(W)$-...
- 19.1k
23
votes
1
answer
995
views
Finite-order self-homeomorphisms of $\mathbf{R}^n$
Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be of finite order if $\phi^m = \mathrm{id}_{\mathbf{R}^n}$ for some ...
- 1,036
22
votes
5
answers
27k
views
What is the standard notation for group action
Please let me know what is the standard notation for group action.
I saw the following three notations for group action.
(All the images obtained as G\acts X for ...
22
votes
2
answers
1k
views
When does $G\times G\times G$ admit a faithful group action on a set of size $|G|$?
[Edited due to YCor's comment:]
Given a finite group $G$, under what conditions does $G\times G\times G$ (the direct product of three copies of $G$) admit a faithful group action on a set of size $|G|$...
- 525
21
votes
1
answer
671
views
Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
- 42.4k
21
votes
1
answer
548
views
Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 19.3k
20
votes
5
answers
3k
views
How to compute the (co)homology of orbit spaces (when the action is not free)?
Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
- 201
19
votes
1
answer
1k
views
A result on Lie group actions on 15-dimensional spheres?
In this interview by Eric Weinstein to Roger Penrose, Timestamp 1:24:05., what result is the host talking about?
Transcription of the relevant part:
"If you have two sets of symmetries, known as ...
- 22.4k
17
votes
1
answer
365
views
Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?
Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma,
$$
|V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}.
$$
Since $g-I$ ...
- 5,514
17
votes
1
answer
493
views
Is a smooth action of a semi-simple Lie group linearizable near a stationary point?
Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
- 15.1k
17
votes
0
answers
420
views
On manifolds which do not admit (smooth) actions of finite groups
Question: Assume a smooth manifold $M$ does not admit any effective smooth group actions of finite groups $G \neq 1$, does it follow that $M$ also admits no continuous effective group actions of ...
- 517
17
votes
0
answers
568
views
Actions on ℍⁿ generated by torsion elements
Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...
- 42.4k
16
votes
3
answers
953
views
Actions on Sⁿ with quotient Sⁿ
What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?
Comments.
I am mostly interested in (maybe trivial) properties of such actions ...
- 42.4k
16
votes
2
answers
599
views
why most of the angles are right
The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which ...
- 42.4k
16
votes
1
answer
914
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 ...
- 4,858
15
votes
10
answers
5k
views
Looking for interesting actions that are not representations
As a person interested in group theory and all things related, I'd like to deepen my knowledge of group actions.
The typical (and indeed the most prominent) example of an action is that of a ...
- 364
15
votes
0
answers
615
views
"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]
My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
- 5,791
14
votes
5
answers
2k
views
A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$
$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...
- 1,000
14
votes
3
answers
506
views
Proving convergence of sum over $\mathbb{Z}^n$
In my research, I am trying to use the following construction by Benson Farb and John Franks, which proves that for all $n$, the group of $n\times n$ matrices with 1's on the diagonal, 0's above the ...
- 143
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 $...
- 5,473
14
votes
1
answer
647
views
If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?
This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$
$\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
14
votes
2
answers
627
views
Action that is Bourbaki proper but not Palais proper
I'm working with different definitions of proper action (Cartan, Bourbaki and Palais) and the relation between them. All the spaces I'm working with are $T_{3.5}$, the definitions are:
If $U$ and $V$ ...
- 193
14
votes
2
answers
1k
views
Symmetric group action on squarefree polynomials
The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest.
Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
- 15.3k
13
votes
2
answers
688
views
Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?
More precisely, is there a criterion that decides the above question?
I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...
- 183
13
votes
3
answers
1k
views
The classifying space of a gauge group
Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = g^{-1}f(p)...
- 801
13
votes
2
answers
600
views
Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$
I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
- 2,616
13
votes
1
answer
690
views
Counterexample showing that G-invariant de Rham cohomology different from cohomology of G-invariant sub-complex?
If $G$ is a discrete or a Lie Group acting smoothly on a manifold $M$, we can define the algebra of $G$-invariant de Rham classes, $H(M)^G$, and we can also consider the cohomology of the sub-complex ...
- 1,326
13
votes
1
answer
998
views
Category without identities?
Just as a monoid is a category with a single object, a semigroup may be seen as a non-unital category, still with associative composition. Then an $S$-set for $S$ a semi-group can be seen as a functor ...
- 10.3k
13
votes
1
answer
1k
views
When taking the fixed points commutes with taking the orbits
Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)
The set $\text{Fix}_H(X)$ of $H$-fixed ...
- 27k
12
votes
1
answer
422
views
Topological amenability vs amenability of an action
Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...
- 121
12
votes
1
answer
1k
views
Cobounded ⇒ cocompact?
Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and
$$\operatorname{diam} H/\Gamma\le 1.$$
Is it true that $H/\Gamma$ is compact?
Stupid example. Assume the action of $\...
- 42.4k
11
votes
2
answers
884
views
Not very transitive actions
Suppose $m$ is a positive integer.
I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-...
- 42.4k
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
11
votes
1
answer
426
views
Scalar curvature and the degree of symmetry
Let $M$ be a closed connected smooth manifold. We define the degree of symmetry of $M$ by $N(M):=\sup_\limits{g}\mathrm{dim}\,\mathrm{Isom}(M,g)$, where $g$ is a smooth Riemannian metric on $M$ and $\...
- 1,699
11
votes
2
answers
2k
views
Determinant associated with a group action
Let $G$ be a finite group and $S$ be a finite set, with $G$ acting on $S$. I consider indeterminates $x_g$ indexed by $g\in G$ and form the matrix of the group action $A\in M_{S\times S}$. Its entries ...
- 50.6k
11
votes
2
answers
1k
views
Orbifolds vs. branched covers
Forgive me if this is a basic question. I'm just learning about orbifolds, and covering spaces are my happy place for thinking about group actions.
If $M$ is a manifold and $G$ is a group acting ...
- 111
11
votes
1
answer
604
views
Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex
Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$.
(Terminology: By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is ...
- 2,064
11
votes
1
answer
524
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
10
votes
3
answers
772
views
Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?
I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!
- 247
10
votes
4
answers
1k
views
When do isometric actions exist?
Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
- 545
10
votes
3
answers
900
views
Regular subsets of $\text{PSL}(2, q)$
Suppose a group $G$ acts on a set $\Omega$. Call a subset $A \subset G$ regular (or sharply transitive or simply transitive or...) if for every two points $\omega_1, \omega_2 \in \Omega$ there is a ...
- 8,903
10
votes
1
answer
1k
views
Białynicki-Birula theory for non-complete varieties
I would like to know to which extent the theory developed for smooth projective varieties in the following articles
A. Białynicki-Birula, Some theorems on actions of algebraic groups.
Ann. of ...
- 22.4k
10
votes
2
answers
603
views
What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$
The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
- 700
10
votes
1
answer
538
views
An equivariant social choice in Mathematical economics
Motivated by this paper and its economics motivations, we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times\cdots\times M}^{\text{$n$ times}}\to ...
- 280
10
votes
1
answer
230
views
A published proof for: the number of labeled $i$-edge ($i \geq 1$) forests on $p^k$ vertices is divisible by $p^k$
Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices (A138464 on the OEIS). The first few values of $F(n;i) \pmod n$ are listed below:
$$\begin{array}{r|rrrrrrrrrrr}
& i=0 &...
- 1,327