Questions tagged [group-actions]
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476
questions
6
votes
1
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Non-Hamiltonian actions in physics
I was reading the following article when I came across the interesting sentence
"non-Hamiltonian [symplectic group] actions also occur in physics"
I took a cursory look at the article cited but ...
21
votes
1
answer
677
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 $...
9
votes
0
answers
340
views
Which polynomials in the minors of a matrix are invariant under conjugation?
$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
0
votes
1
answer
240
views
Measure on group invariant under group action on metric space
This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO.
The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...
3
votes
0
answers
76
views
Circle actions on simply connected spin manifolds
Recently I've been stuck by a concrete problem. I'll try to make it more general.
Suppose $M$ is a simply connected spin manifold (with higher enough dimension), and $S^1$ acts on $M$ effectively. ...
3
votes
1
answer
205
views
Operation of a p'-group on a set of p-power order and fix points
The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....
5
votes
1
answer
108
views
(Euclidean) open orbit in an irreducible real algebraic set
Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...
1
vote
1
answer
243
views
Understanding a group of transformations of the plane $\mathbb{Z} \times \mathbb{Z}$
I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...
8
votes
2
answers
412
views
Torus action implying infinite fundamental group
Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?
Consider the ...
1
vote
0
answers
72
views
Volume form preserved by the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $?
I know this is quite an elementary question but I am not an expert in Lie theory.
Does the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $ ...
1
vote
0
answers
162
views
sequence definition of proper group action
My understanding is that for an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the map $G \times M \rightarrow M \...
5
votes
1
answer
245
views
Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?
A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...
8
votes
1
answer
250
views
$\mathbb{C}^{*}$-actions on Fano $3$-folds
I am looking for an example of a smooth Fano $3$-fold $X$ over $\mathbb{C}$, with a non-trival $\mathbb{C}^{*}$-action, which satisfies the following properties:
There is a $\mathbb{C}^{*}$-action ...
2
votes
0
answers
92
views
Effective actions by non-commutative groups have non-commuting fundamental vector fields?
I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...
0
votes
1
answer
247
views
Ordered group acting freely on partially ordered set
Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering:
$$ s_1 &...
1
vote
0
answers
213
views
A question concerning a short exact sequence with an action
Let $A$ and $D$ be two non-trivial abelian groups and $B,C$ be two non-abelian groups. Also, let $C$ is a free group and acts on $A,B,D$.
Let $0\to A \xrightarrow{f}B\xrightarrow{g}C\to 0$ be a ...
2
votes
0
answers
74
views
Does this condition imply symplectic birational cobordism?
From the definition of symplectic birationality given here (https://arxiv.org/pdf/0906.3265.pdf, Definition 2.1), two compact symplectic $2n$-manifolds $(M_{1},\omega_{1}),(M_{2},\omega_{2})$ are ...
3
votes
1
answer
360
views
Unclear construction in a paper of Ornstein and Weiss
I originally posted this on math.stack, but no one answered, so im posting here:
I need help understanding the following construction (Taken from the paper "Entropy and isomorphism theorems for ...
14
votes
3
answers
519
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 ...
6
votes
0
answers
372
views
closed substack of quotient stack
The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...
1
vote
0
answers
109
views
Torus action on hypersurface of multidegree $(1,1,1,1)$ in $(\mathbb{CP}^{1})^{4}$
Let $X$ be a (smooth) hypersurface of mulidegree $(1,1,1,1)$ in $(\mathbb{CP}^{1})^{4}$. According to table $6$ (page 19) in the following paper https://arxiv.org/pdf/1508.01089.pdf, we have:
$X$ has ...
13
votes
1
answer
700
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 ...
7
votes
1
answer
582
views
Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
3
votes
1
answer
474
views
Orbits of unipotent groups over local fields are closed?
Let $H$ be a connected, unipotent linear algebraic group defined over a local field $k$. Let $H \times_k X \rightarrow X$ be an action of $H$ on an irreducible, affine $k$-variety $X$ which is ...
4
votes
1
answer
104
views
Sufficient conditions for secondary invariants
Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...
5
votes
0
answers
207
views
Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow
Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
3
votes
1
answer
431
views
Is there an easy example of group action where the slice theorem produces a non-trivial principal bundle?
Let $\rho$ be a group action by a compact group $G$
\begin{equation}
\rho:G\times M \rightarrow M \\
\rho:(g,p) \rightarrow \rho_g(p)
\end{equation}
Denote the orbit of $p\in M$ by $\...
14
votes
2
answers
636
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$ ...
6
votes
0
answers
227
views
Recovering SU(2)-space when the orbit space is a 3 sphere with 3 singular orbits
Background: Consider $SU(2)$ action on the 6-dim flag manifold $M=SU(3)/T^2$ via left multiplication. We view $SU(2)$ as a subgroup of $SU(3)$ corresponding to $2\times 2$-block. The action is just ...
0
votes
0
answers
234
views
Quotient by augmentation ideal
Let $p$ be a prime number. Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers and let $R = \mathbb{Z}_p [[X_1, \ldots, X_n]] / (f_1, \ldots, f_d)$.
Assume that a finite abelian group $G$ of order ...
6
votes
0
answers
137
views
Existence of $G$-map between finite $G$-simplicial complex
Let $X, Y$ be finite free $G$- simplicial complex. What kind of properties are necessary for existence a $G$-map,i.e, a continuous map which preserves $G$-action, from $X$ to $Y$? Does existence of ...
0
votes
1
answer
140
views
Dualizing the trivial action on a $C^*$-algebra
Let $G$ be a finite abelian group (cosidered as a discrete topological group), $A$ a unital separable $C^*$-algebra. Let $T\colon G\to \operatorname{Aut}(A)$, $T_g(a)=a$ for all $g\in G$ the trivial ...
2
votes
0
answers
41
views
On the minimum distance along an orbit
Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define
$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...
1
vote
0
answers
191
views
free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$
I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.
For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 ,...
5
votes
1
answer
431
views
Action of upper triangular matrices
Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$.
Do there exist two upper triangular matrices ...
2
votes
1
answer
122
views
Hamiltonian Group action with infinitely many stabiliser types
What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that ...
3
votes
0
answers
584
views
Group acting freely on tree
A tree is a connected acyclic (symmetric) graph. A group acts freely on a graph if there are no inversion of edges and stabilizers of vertices are trivial.
The Bass-Serre Theorem states that A group ...
2
votes
1
answer
395
views
Action on a normal subgroup where each coset acts freely
Given a group G and a normal subgroup N of G, is there an action of G on N such that, whenever g,h are distinct members of the same N-coset, we have g•n≠h•n? If not, then can this be done in the case ...
9
votes
1
answer
803
views
6-manifolds admitting SO(3) action with 2 orbit types
Let $M^6$ be a 6-dimensional smooth manifold, on which the group $G=SO(3)$ acts smoothly with 2 orbit types $SO(3)/SO(2)$ and $SO(3)$, such that the orbit space $X=M/SO(3)$ is a 3-ball $B^3$, whose ...
1
vote
0
answers
329
views
Does a quotient group $G/N$ have a natural action on the regular representation of $G$?
Let $G$ be a group. I am happy to assume niceties such as finite and abelian, but perhaps it is not necessary to answer my question.
Consider the $|G|$-dimensional vector space $V$ (over some nice ...
3
votes
2
answers
296
views
Affine connections as equivariant maps
An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^*M)\otimes TM$. One would think that this affine ...
0
votes
1
answer
504
views
number of orbits of a proper subgroup
Let $G$ be a permutation group that acts on (say) $X=\{1,2,..,n\}$, and $H$ be a proper subgroup of $G$. Can one say anything precise about when the number of orbits of $H$ on $X$ will be equal to ...
24
votes
1
answer
1k
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 ...
4
votes
1
answer
599
views
Fixed point of a group action
Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...
11
votes
1
answer
625
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 ...
4
votes
1
answer
235
views
Is the class (resp. Picard) group of a $G$-variety generated by invariant divisors?
Let's work over the complex numbers. Let $S$ be a normal surface, $\mathrm{A}^1(S)$ the class group of divisors on $S$ and $\mathrm{Pic}(S)$ its Picard group. Let $G$ be a reductive group acting on $S$...
5
votes
0
answers
412
views
Infinity categories with an action of a simplicial group
Recent papers in derived algebraic geometry use a notion of $S^1$-actions on infinity categories. I think I understand what this "should" be and how to calculate with it; however, I can't find a much ...
1
vote
0
answers
311
views
Fixed point set of torus action is discrete and infinite?
Let $T=(\mathbb{C}^*)^k$ act holomorphically on a smooth quasi-projective complex algebraic variety $M$.
Can the fixed point set $M^T$ be discrete and infinite?
I think the answer is no because ...
4
votes
0
answers
899
views
Contracted product of torsors
Given a group $G$ and a left $G$-set $X$, then we can make $X$ a right $G$-set defining the action as $xg:=g^{-1}x$ , or if you prefer we are considering the opposite group $G^{op}$ to make the left ...
14
votes
1
answer
657
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 ...