# Questions tagged [group-actions]

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### Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon?

When I say that an $r$-sided simple (i.e., not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular ...
569 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 ...
• 10.6k
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### Question about finite dimensional representations of a semi-simple Lie group

I have posted a question in MSE https://math.stackexchange.com/questions/4468138/question-about-finite-dimensional-representations-of-a-semi-simple-lie-group but didn't receive any comment or answer. ...
• 53
166 views

### Special cell decomposition for spheres with free $\mathbb{Z}/p\mathbb{Z}$-action by orthogonal transformations?

Consider the unit sphere $S^d$ in $\mathbb{R}^{d+1}$ with the antipodal action $\nu \colon x\mapsto -x$. This turns $S^d$ into a free $\mathbb{Z}/2\mathbb{Z}$-space. Construct a CW-complex structure ...
• 43
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### Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...
• 3,399
1 vote
45 views

### Example on pseudo-groups

Definition: ‎A pseudo-group is a collection $\mathcal{G}$ of (locally defined) invertible smooth diffeomorphisms of a manifold $M$. ‎The simplest example of a pseudo-group is the collection of all ...
251 views

### On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$. Is it true ...
• 19.3k
1 vote
24 views

• 195
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### Do there exist methods for determining the orbits of a group action on the cartesian product of sets?

Suppose that we have some group $G$ acting on some set $\Omega$. Then $G$ acts on $\Omega^n = \Omega \times \cdots \times \Omega$ ($n$ times) naturally. I wonder, is there an iterative algorithm to ...
• 161
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### 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 $|G|$...
• 433
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• 487
1 vote
116 views

### Equivariant embeddings

According to the article I got from this question, closed Riemann manifolds together with their isometry groups (which is always a compact Lie group) can be smoothly and equivariently embedded into ...
• 710
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### Fundamental domains for proper Lie group actions on smooth manifolds

The setting: $M$ an arbitrary smooth manifold, $G$ a Lie group acting effectively and properly on $M$ by diffeomorphisms. Motivation: when trying to figure out the homeomorphism type of the orbit ...
• 123
114 views

### The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...
• 73
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• 4,819
1 vote
158 views

### Symplectic structure on a vector bundle

Let $G$ be a Lie group and let $M \rightarrow B$ be a $G$-equivariant vector bundle with typical fiber $E$. Suppose that $B$ and $E$ are both symplectic manifolds with a $G$-Hamiltonian action. Can we ...
• 53
153 views

### Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...
• 113
1 vote
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• 1,639
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### Equivalent definitions of strongly proximal action

Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar, Kennedy and Ozawa: I have two questions: (1) What ...
• 405
235 views

### Behavior of canonical divisor under a finite group quotient

Given a smooth algebraic surface $X$, and a group $G$ acting on it and letting $Y := X / G$, how can we compute $K_Y^2$ from from $K_X^2$? Current progress: In Borisov and Fatighenti - New explicit ...
260 views

### Sections of a polar action are totally geodesic

This question is a repost of the following: https://math.stackexchange.com/questions/4195805/sections-of-a-polar-action-are-totally-geodesic. I've decided it to post it here because it didn't seem to ...
1 vote
52 views

### Row-wise conjugation of completely bounded map by group action

Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...
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Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...