Questions tagged [group-actions]

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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 ...
R Mary's user avatar
  • 949
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 $...
Anton Petrunin's user avatar
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 ...
Asaf Shachar's user avatar
  • 6,611
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 ...
Sellerie's user avatar
  • 103
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. ...
Ivy's user avatar
  • 123
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....
Sven Wirsing's user avatar
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 ...
AleAlvAlwaysDIEZ's user avatar
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 $\...
Hans-Peter Stricker's user avatar
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 ...
Lawrence Mouillé's user avatar
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}) $ ...
Selim G's user avatar
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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 \...
X-Naut PhD's user avatar
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 $B\subset G$ ...
Taras Banakh's user avatar
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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 ...
Nick L's user avatar
  • 6,923
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 $G$ ...
R Mary's user avatar
  • 949
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 &...
lunchmeat's user avatar
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 ...
MHenry's user avatar
  • 139
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 ...
Nick L's user avatar
  • 6,923
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 ...
Alfred's user avatar
  • 31
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 ...
P. May's user avatar
  • 143
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 ...
aytio's user avatar
  • 371
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 ...
Nick L's user avatar
  • 6,923
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 ...
ychemama's user avatar
  • 1,326
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 ...
THC's user avatar
  • 4,313
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 ...
D_S's user avatar
  • 6,100
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 ...
Jared's user avatar
  • 778
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 ...
Ali Taghavi's user avatar
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 $\...
R Mary's user avatar
  • 949
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 $V$ ...
Marcos TV's user avatar
  • 193
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 ...
Yuhang Liu's user avatar
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 ...
JWM's user avatar
  • 183
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 ...
MathFun's user avatar
  • 233
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 ...
Sabrina Gemsa's user avatar
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)...
Eduardo Longa's user avatar
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 ,...
Shivani Sengupta's user avatar
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 ...
user avatar
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 ...
Nick L's user avatar
  • 6,923
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 ...
poset's user avatar
  • 63
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 ...
Michael Cotton's user avatar
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 ...
Yuhang Liu's user avatar
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 ...
Ruben Verresen's user avatar
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 ...
user avatar
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 ...
vgmath's user avatar
  • 133
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 ...
Wille Liu's user avatar
  • 1,056
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 ...
Anton Petrunin's user avatar
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 ...
Michał Kukieła's user avatar
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 $S$...
Qfwfq's user avatar
  • 22.7k
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 ...
Harrison Chen's user avatar
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 ...
HLC's user avatar
  • 287
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 ...
HaroldF's user avatar
  • 433
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 ...
Anthony Carapetis's user avatar

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