Questions tagged [group-actions]
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37
questions
31
votes
7
answers
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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 $\mathfrak{S}_n$ ...
14
votes
5
answers
2k
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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 ...
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 ...
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 ...
7
votes
1
answer
262
views
Is $\mathbb{Q}$ "equivalent" to a structure with transitive automorphism group action?
Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For ...
4
votes
2
answers
883
views
Orbits of a symplectic group on its Lie algebra in the finite field case
The classical problem regarding the action of symplectic group on its Lie algebra gives rise to the following question in the finite field case.
Let $\mathbb F_p$ be a finite field. Then the ...
1
vote
1
answer
148
views
'Convex' slices of proper actions
Consider a Lie group $G$ acting properly on a manifold $M$. Then by the slice theorem we can find for any point $m\in M$ a submanifold transverse to the orbit $\mathcal{O}$ through $m$ and which is (...
1
vote
1
answer
226
views
locally closed orbits in metric Hausdorff topology
I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that
Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
23
votes
5
answers
28k
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 ...
21
votes
1
answer
550
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}(\...
16
votes
1
answer
945
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 ...
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 $...
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 ...
12
votes
1
answer
470
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 ...
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 ...
9
votes
1
answer
302
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Counterexamples to analogue of Cannon conjecture in higher dimensions
It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for $\mathbb{H}^...
9
votes
2
answers
510
views
Quotients of schemes by connected groups
Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.
By the Keel-Mori theorem, the quotient $X/G$ is ...
7
votes
1
answer
273
views
Question about an example in symplectic geometry
Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
7
votes
2
answers
865
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Is there a topological Chevalley-Shephard-Todd Theorem?
Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( ...
6
votes
1
answer
293
views
Trans-amenability of group actions
This problem is derived from this post.
Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are ...
6
votes
2
answers
559
views
Stabilisers of group actions
Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.
Suppose there exists some point $x \in X$ whose stabiliser ...
6
votes
0
answers
221
views
A group action on another group action quotient: how to best describe the resulting structure and does it have a name?
Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits.
Is there a nice ...
5
votes
1
answer
268
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 ...
5
votes
1
answer
578
views
Fixed points on boundary of hyperbolic group
Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?
5
votes
1
answer
840
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A generalized Burnside's lemma
Let $G$ be a finite group acting on a set $X$, and let $S\subseteq G$ be a union of conjugacy classes. Then I believe I can prove:
$$ \sum_{[x]\in X/G} \frac{|G_x \cap S|}{|G_x|} = \sum_{g\in S} \...
4
votes
1
answer
247
views
Homotopy type of G-CW-structure
Let $G$ be a finite group, and $X$ be a free $G$-space. Moreover, assume that $X$ has a homotopy type of a CW-complex. Does $X$ have $G$-homotopy type of a $G$-CW complex also?
Edit: My main ...
4
votes
2
answers
812
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Künneth formula for Bredon cohomology theory
Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...
3
votes
1
answer
1k
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cohomology of the orbit space of a group action
Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...
3
votes
2
answers
296
views
Affine connections as equivariant maps
An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^*M)\otimes TM$. One would think that this affine ...
3
votes
0
answers
241
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Simultaneous coset spaces [closed]
Let $X$ and $Y$ be sets. Under what conditions is there a group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in $Y$?...
3
votes
2
answers
360
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Is this a submanifold?
Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by:
$$\psi : G \...
3
votes
0
answers
122
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A topological space extracting from a group action
Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to \mathbb{C}$...
2
votes
2
answers
338
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A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$
Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}_n(k)$ $p$-nilpotent if $X^p=0$, and let $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k))$ be the set of ...
2
votes
0
answers
244
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A possible generalization of the Borsuk Ulam theorem via action of symmetric groups
The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...
2
votes
0
answers
303
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Rational conjugation of elements of a finite group
Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...
1
vote
0
answers
182
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Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix
Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
0
votes
0
answers
249
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A question concerning some group action
Let $G$ be a finite group. Consider the set
$$X = \bigcup_{H \le G} G/H$$
which is a disjoint union of left cosets of subgroups $H$ of $G$.
Then $G$ acts on $X$ by left multiplication, and the number $...