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7
votes
2answers
103 views

Is there a subset of $\Sigma_n$ s.t. each pair of elements is once in each pair of positions?

Is there a subset $A \subset \Sigma_n$ such that for each pair $(x, y)$ and each pair $(i, j)$, there is exactly one permutation $\sigma \in A$ such that $\sigma(i) = x$ and $\sigma(j) = y$? Remark ...
3
votes
0answers
58 views

Principal orbits for hamiltonian actions

Let $G$ be a compact Lie group which acts by symplectomorphisms on a symplectic manifold $(M,\omega)$. Futhermore let $\mu \colon M \to \mathfrak g$ be a moment map for this action. Denote by $\Omega ...
6
votes
1answer
136 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
3
votes
0answers
50 views

Transverse intersection in the $G$-orbit of paths

I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it? Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...
3
votes
2answers
85 views

Fixed point set for a subcircle of torus actions

Let $T=S^{1}\times S^{1}\times ...\times S^{1}$ ($n$ times) be $n$ dimensional torus and $X$ be a $T$-space. Lemma: If $X$ has finitely many connected orbit type, then there is a subcircle ...
2
votes
1answer
135 views

isometric action on the $n$-sphere

Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely $$ \Psi \colon S^1 \times ...
4
votes
1answer
224 views

Is the direct sum in Maschke's Theorem an orthogonal decomposition?

I am reading a paper on coding theory, and it uses a statement, which was claimed to be a reformulation of Maschke's Theorem. But I felt that was false... Let's say $\mathcal(V):=\mathcal{F}_2^n$ is ...
4
votes
1answer
68 views

Orientability of orbit type strata of Lie group actions

Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...
3
votes
1answer
56 views

vector bundles induced by an action of a finite subgroup of $O(n)$

Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle $$ \xi(M,G): \mathbb{R}^n\longrightarrow ...
3
votes
2answers
112 views

Extension of a group action beyond the boundary

Let $M$ be a compact manifold with boundary and suppose a compact group $G$ acts on it. Can one always extend the action beyond the boundary? More precisely, does there always exist a $G$-manifold ...
0
votes
1answer
169 views

group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram If $\xi$ is a trivial bundle, ...
10
votes
3answers
426 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 ...
2
votes
0answers
167 views

Manifolds as simultaneous coset spaces

Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie 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 ...
3
votes
0answers
203 views

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 ...
14
votes
1answer
371 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 ...
3
votes
1answer
130 views

How to construct a proper action of a group of finite virtual cohomological dimension?

Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$, where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual ...
0
votes
0answers
55 views

Submanifolds invariant under subgroups with identical quotients

given a smooth manifold $M^n$ and a finite group $G$ acting smoothly and effectively, let's consider two (embedded) $k$-dimensional submanifolds $N_1,N_2\subset M$ and two subgroups $H_1,H_2\subset G$ ...
3
votes
1answer
109 views

The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$

Motivated by the following RG question we ask a related question as follows: We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes ...
3
votes
2answers
134 views

equivariant embeddings from the k-th configuration space to the k+1-th configuration space

Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
1
vote
1answer
143 views

Averaging measurable functions over amenable group actions

Let $G$ be an amenable group acting on a space $X$. Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$. Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
3
votes
1answer
118 views

On an example by Romagny about fixed point stack not commuting with coarse moduli space

This is to understand better Example 3.9 on page 221 of Group actions on stacks and applications by M.Romagny. For an action of an algebraic group (scheme) $G$ on an algebraic stack $\mathcal{M}$, ...
2
votes
2answers
118 views

Invariant polynomials under the action of $H\le\operatorname{GL}_n(\mathbb{F}_p)$

Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}_n(\mathbb{F}_p)$ acts on the polynomial ring $\mathbb{F}_p[x_1,\ldots,x_n]$ via $A\cdot x_i=\sum_j a_{ji}x_j$ for ...
5
votes
0answers
98 views

On finite quotients of unions of $n$ affine varieties

Assume that a finite group $G$ acts on a quasi-projective variety $Q$ (say, over complex numbers) that possesses a Zariski cover by $\le n$ affine varieties. My question is: does the quotient $Q/G$ ...
9
votes
2answers
453 views

Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of ...
6
votes
2answers
342 views

Algebraic proof without using comparison theorem for étale cohomology

Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity). The étale cohomology groups of X are therefore equipped with an action of ...
4
votes
1answer
126 views

Is this notion of 'closed subset' of a semigroup action something people have thought of?

Suppose $S$ is a semigroup (or a monoid, or a category), and $X$ is an $S$-set -- that is, a set with an action of $S$. Say that a sub-$S$-set $Y$ is "downward closed" (or maybe "well-generated") if ...
6
votes
2answers
213 views

“Interesting” projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?

The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would ...
3
votes
1answer
114 views

symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example, $\Delta^2$: $\Delta^3$: Let ...
12
votes
3answers
459 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 ...
2
votes
1answer
180 views

Kunneth formula of Cartesian product modulo orders of coordinates

Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on $$ \prod_n X $$ by $$ \sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in ...
1
vote
1answer
275 views

Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?

$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times. I am studying some function arising from symplectic geometry which happens in my case to be ...
4
votes
1answer
290 views

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 ...
6
votes
2answers
642 views

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. ( ...
2
votes
1answer
166 views

Actions of the unit circle on finite complex matrices

Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle. How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on ...
3
votes
1answer
278 views

Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.) Suppose we have an algebraic group $G$ ...
3
votes
1answer
291 views

simultaneous action of GL(n) on the matrices

Consider the action of GL(n,k) on the set MxM where M is the set of all n-by-n matrices over k given by $g.(h,l) \mapsto (ghg^{-1}, glg^t)$. Individually these actions are well studied and good ...
1
vote
0answers
106 views

Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$

I ask this question in MSE and I received no answer, so I repeat it here: Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...
4
votes
0answers
195 views

Can Z/2 x Z/2 act freely on an infinite dimensional sphere?

Using that all groups that act freely on some sphere $S^n$ have periodic cohomology, one can see that $\mathbb Z/2 \times \mathbb Z/2$ can not act freely on any $S^n$. But can it act freely on ...
6
votes
1answer
219 views

Free actions of non-amenable groups

Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free ...
6
votes
1answer
138 views

Example of a $G$-sphere that is not a $G$-representation sphere

Let $G$ be a finite group with the discrete topology. To set terminology: a $G$-sphere is a sphere equipped with a continuous $G$-action a $G$-representation sphere is a $G$-sphere obtained from an ...
0
votes
0answers
114 views

Action of semidirect products of cyclic groups

Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...
6
votes
2answers
287 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 ...
22
votes
4answers
1k views

Dividing by two in the category of vector spaces

Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$? Here "naturally" means "in an $GL(V) \times ...
2
votes
1answer
258 views

finite stabilizers + compact orbit space => proper action?

Suppose a countable discrete group is acting on a smooth manifold with finite stabilizers and the orbit space is compact (and Hausdorff). How one can prove that the action is proper? I found this ...
1
vote
0answers
60 views

Name for condition on map of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that $k(\epsilon)=\epsilon$ for all $a,b\in M$, there exists $v\in N$ such ...
1
vote
1answer
141 views

The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...
1
vote
2answers
291 views

Connected components of algebraic groups

Let $G$ be an algebraic group, and $G_{Id}$ the connected component of the identity. Then $G_{Id}$ is a normal subgroup of $G$ and $G/G_{Id}$ is the component group of $G$. Let $G_{c}\subset G$ be ...
1
vote
1answer
257 views

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 ...
5
votes
2answers
260 views

fixpoint algebras of a permutation action

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...
6
votes
0answers
180 views

Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?

Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...