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0
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1answer
77 views

Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$

For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$. For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...
0
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0answers
99 views

What insight does a regular group-action of a linear algebraic group on an algebraic variety give about the isolated points of the latter?

I'm trying to gather clues on solving this bigger problem (MO link). So, say I have an algebraic variety $\Omega$ over a field $F$ (e.g $\mathbb R$, $\mathbb C$, $\mathbb F_2$, $\mathbb F_q$). ...
5
votes
1answer
103 views

Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
2
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0answers
93 views

Projective G-group

Let $G$ be a fixed group. Can there be projective $G$-groups which are not free $G$-groups? If yes, for which groups $G$ it happens? By a "projective $G$-group", I mean a projective object in the ...
5
votes
1answer
164 views

Nielsen-Schreier with operations

The Nielsen-Schreier theorem states that subgroups of a free subgroup are free. Is this hold also for groups with operations? Explicitly, let $G$ be a fixed group. Let $F$ be a group with $G$-action ...
1
vote
2answers
97 views

Computational complexity of sizes and number of orbits of a group acting on a set

I'm interested in the relation between the computational complexity of counting orbits and counting elements in orbits for groups acting on sets. More formally: Assume that $X_n$ is a infinite ...
2
votes
1answer
115 views

Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space

Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous ...
4
votes
1answer
86 views

Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism

I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...
5
votes
1answer
167 views

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 ...
19
votes
1answer
532 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 $...
6
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0answers
124 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
1answer
74 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
0answers
54 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
1answer
87 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
1answer
75 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 ...
2
votes
1answer
205 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
2answers
300 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
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0answers
64 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
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0answers
50 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
1answer
190 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
1answer
204 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
0answers
59 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
1answer
97 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 &...
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0answers
107 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 ...
1
vote
0answers
52 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
1answer
266 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
3answers
371 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 ...
5
votes
0answers
139 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
0answers
87 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 ...
10
votes
1answer
380 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 ...
6
votes
1answer
401 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 ...
1
vote
1answer
137 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
1answer
70 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
0answers
109 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
1answer
136 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 $\...
13
votes
2answers
428 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 $V$ ...
5
votes
0answers
168 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
0answers
104 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
0answers
120 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
0answers
117 views

Irreducibility of fixed point scheme

Let $X$ be an integral affine scheme $X = Spec(A)$ endowed with a finite groupe action by $G$, which is of order $n$. Consider the fixed points scheme $X^G$ Assume $n$ is invertible in $A$. Is ...
1
vote
1answer
101 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
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0answers
33 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
0answers
143 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 ,...
6
votes
1answer
334 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
1answer
105 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
0answers
169 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
1answer
235 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 ...
10
votes
1answer
471 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
0answers
150 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
2answers
170 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 ...