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Questions tagged [equivariant-maps]

Questions about or involving equivariant maps, the natural maps between $G$-sets.

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Is the change of trivialisations on a principal $G$-bundle given above each basepoint by a right-multiplication with a group element?

Let $\pi: M \rightarrow B$ be a principal $G$-bundle. Suppose $(U_i, \phi_i)$, $(U_j, \phi_j)$ are two (equivariant) trivialisations $\phi_i: \pi^{-1}(U_i) \rightarrow U_i \times G$ (similarly for $\...
rosecabbage's user avatar
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Primitive non-central idempotents of a group algebra

Let $W_i;\ 1 \le i \le m$ be irreducible $\mathbb{C}$-representations of a finite group $G$ with $\mathbb{C}$-characters $\chi_i$. Let $(V,\rho$) be a $\mathbb{C}$-representation of $G$ with isotypic ...
khashayar's user avatar
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3 votes
1 answer
50 views

Intertwiner space isomorphism with invariante space

Let $V,W$ be two representation vector spaces of some group $G$. Thus, an intertwiner is as a linear map $T:V\rightarrow W$ satisfying $$ T(g\cdot v)=g\cdot T(v) $$ We'll denote the space of ...
Powder's user avatar
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1 vote
0 answers
18 views

Covariant faithful representations

It is well known that every C*-algebra admits a faithful representation into bounded operators on some Hilbert space. But, what happens in the equivariant world? That is, given a topological group (...
mathable's user avatar
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112 views

A $\mathbb{Z}_2$-equivariant map from $n$-torus to $2$-sphere that is null-homotopic is $\mathbb{Z}_2$-homotopic to a non-surjective map?

I have been thinking on the problem below for a while and I am not sure if it is correct or not. I am trying to see if there exists a counter-example for the problem below. Problem: Let $f: (S^1)^n \...
Arash's user avatar
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4 votes
1 answer
69 views

Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, ...
Fabio Neugebauer's user avatar
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1 answer
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isomorphic groups that are G-equivariant [closed]

Start with two finite groups $A$ and $B$ and a group isomorphism $f$ between them. Let a finite group $G$ act on both $A$ and $B$. By definition $f$ is $G$-equivariant if $g(f(a))=f(g(a))$. Do I ...
tess's user avatar
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2 votes
0 answers
72 views

Confusion about induced bundles, equivariant Vector bundles and representations

I am writing this question to try to unconfuse myself in two different directions. Throughout we work over $\mathbb{C}$ and the choice of topology should not matter, but say we are using the etale ...
Jabberwocky's user avatar
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1 answer
31 views

About equivariant vector valued forms on principal bundle

Let $\pi: M \to E$ be a $G$-equivariant vector bundle and let us adopt the notation $$C^{\infty}(M,E)^{G} = \{\tilde{s} \in C^{\infty}(M,E) | \ \forall g \in G \ \tilde{s} \cdot g = \tilde{s}\}$$ ...
Integral fan's user avatar
1 vote
1 answer
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Existence of a transitive smooth G-action of $G\times G$

This question arose from an attempt of using the Equivariant Rank Theorem to prove question 7.1 of Lee's Introduction to Smooth Manifolds. The Equivariant Rank Theorem is stated: Let M and N be ...
PotusOtis's user avatar
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Decomposition of equivariant maps with symmetry-adapted basis

Let $(V, \rho)$ be a representation of a finite group $G$ whose irreducible representations over complex numbers are $(W_i, \pi_i)$ for $1 \le i \le m$. Suppose $\dim(\text{Hom}_G(W_i),V)=d_i$ and $\...
khashayar's user avatar
  • 2,331
0 votes
1 answer
114 views

Space of equivariant homomorphism from the space of G-linear maps to a vector space

Let $(V, \rho)$ be a representation of a finite group $G$ whose irreducible representations over complex numbers are $(W_i, \pi_i)$ for $1 \le i \le m$. Let $$H_i=\text{Hom}_G(W_i,V)= \{\tau \in \text{...
khashayar's user avatar
  • 2,331
1 vote
0 answers
55 views

Transport of group actions via homeomorphisms

Let $G$ be a topological group acting on the topological space $X$ and let $\phi\colon X\xrightarrow{\simeq} Y$ be a homeomorphism between the spaces $X$ and $Y$. Can I induce a $G$-action on $Y$ by $\...
Mathematics enthusiast's user avatar
2 votes
1 answer
76 views

Equivariant splitting of short exact sequences with a $\mathbb {Z}/2\mathbb{Z}$-action

Let $G = \mathbb{Z}/2\mathbb{Z}$, and consider a short exact sequence of \emph{free} $\mathbb Z$-modules of finite rank endowed with a $G$-action $$ 0\to \mathbb Z^n \to \mathbb Z^m \to \mathbb Z^k\to ...
Overflowian's user avatar
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3 votes
1 answer
104 views

Equivalent definitions of group action, isomorphism of GSet categories

Context: For group actions on a set, there are two habitual definitions and a bijection between objects from each definition: A function $G\times A \rightarrow A$ with the usual properties A group ...
Absent mind's user avatar
1 vote
0 answers
24 views

Nontrivial Monodromy of the Universal Stiefel Bundle (and $O(n)$-equivariant vector fields on spheres)

Note: I'm not allowed to embed images into my posts yet, so I've linked my diagrams instead. Throughout, we will make use of the following result. Fact. For $H$ a Lie subgroup of $G$, there is a ...
Baylee Schutte's user avatar
1 vote
2 answers
154 views

Existence of continuous map from $\mathbb{R}^n$ to $\mathbb{R}^{n-1}$ that "respects norms"

Is there a continuous map from $\pi: \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ that is $(\sim_n, \sim_{n-1})$-invariant where $\sim_n$ is the equivalence relation of equality up to orthogonal ...
gordta_chichrron's user avatar
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0 answers
32 views

Equivariant Loomis-Sikorski-Stone reprsentation theorem

Let $X$ be a set, $\mathcal{B}$ a $\sigma$-algebra on $X$ and $\mu$ a probability measure on $(X,\mathcal{B})$. Suppose $\mathcal{C}$ is a $\sigma$-subalgebra of $\mathcal{B}$. Then, the Loomis-...
Henrique Augusto Souza's user avatar
5 votes
1 answer
107 views

Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces

Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
Mathematics enthusiast's user avatar
3 votes
1 answer
70 views

Existence of a principal bundle charts compatible with $f$-equivariant reductions

Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, and $f:G\rightarrow H$ be Lie group homomorphism. Let $F$ be a principal bundle homomorphism, that is ...
Chris's user avatar
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1 vote
0 answers
73 views

Equivariant Submersion Theorem

I have a question about the Equivariant Submersion Theorem (Proposition 2.7) in the article Equivariant control data and neighborhood deformation retractions by Markus J. Pflaum and Graeme Wilkin. The ...
Bastiaan Cnossen's user avatar
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0 answers
44 views

Two group cohomology are isomorphic to each other if they have the same coefficient.

Suppose $M$ and $N$ are $\mathbb{F}[G]$-modules over a field $\mathbb{F}$, and $G$ is a finite group. Claim: Suppose there is a map $f: M \rightarrow N$ such that $f$ is a $\mathbb{F}$-vector space ...
Rookiecookie's user avatar
1 vote
1 answer
64 views

Why are transition maps of principal bundles given by multiplications of elements of the group?

It seems to be a common fact that transition maps of principal bundles are given by multiplication by a group element (in this post,for example.) The setup is this: Consider a $G$-bundle $E \overset{\...
Tanny Sieben's user avatar
  • 2,471
1 vote
1 answer
71 views

Difficulty with converting Yoneda's natural isomorphy into a group isomorphism in the proof of Cayley's theorem

$\newcommand{\A}{\mathscr{A}}\newcommand{\Gc}{\mathscr{G}}\newcommand{\G}{\mathcal{G}}\newcommand{\s}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sym}{\mathsf{Sym}}$I am having ...
FShrike's user avatar
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1 vote
0 answers
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Equivariant maps up to homotopy

Let $G$ be a topological group and $X$, $Y$ be $G$-spaces. Suppose there is a continuous map $F:X\to Y$ which is $G$-equivariant up to homotopy, namely the two maps $G\times X\to Y$, $(g,x)\mapsto F(...
Yeah's user avatar
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2 votes
1 answer
192 views

G-equivariant automorphisms of G

Let $G$ be group, and $$ \operatorname{Aut}_G(G):=\{f\in\operatorname{Aut}G\mid f(xg)=f(x)g,\forall x,g\in G\} $$ I want to show that in fact $\operatorname{Aut}_G(G)\cong G$. $G\subset \operatorname{...
user avatar
2 votes
1 answer
68 views

Do equivariant dynamics have invariant equilibria?

Let $\phi_t$ be a one-parameter subgroup of diffeomorphisms of a manifold ($\mathbb{R}^n$ for simplicity). In other words, $\varphi$ is a continuous dynamics. Suppose that $\varphi_t$ is $G$-...
gm01's user avatar
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1 vote
1 answer
426 views

Equivariant map between two vector spaces definition and formulation.

Reading the following paper (Proposition 2.2) https://arxiv.org/pdf/1804.10306.pdf I'm stuck trying to understand the following: We have a compact group $\Gamma$, $V,U$ two vector spaces carrying a ...
James Arten's user avatar
  • 1,953
2 votes
1 answer
50 views

A question about extending group actions to maps.

It is known that an action $\alpha: G \times M \rightarrow M$ induces for each group element $g \in G$ a transformation on $M$, $T_g : M \rightarrow M.$ Now what is done to extend this action to $\...
James Arten's user avatar
  • 1,953
-1 votes
1 answer
30 views

Does every layer of an equivariant composite function have to be equivariant?

For example, given a composite function $f(x)=f_1(f_2(f_3(x)))$, if $f(x)$ is equivariant to the group $G$ (e.g. $SO(3)$), then is it necessary for $f_1$,$f_2$,$f_3$ to be equivariant to $G$? Thanks.
Alice Yang's user avatar
1 vote
1 answer
48 views

A question about equivariance to 3D transformations using semi-direct and direct products.

in the paper: https://arxiv.org/pdf/2010.02449.pdf the author consider some classes of functions that are invariant to the action of the group $G= \mathbb{R}^3 \rtimes SO(3) \times S_n$ on $\mathbb{R}^...
James Arten's user avatar
  • 1,953
2 votes
2 answers
250 views

Equivariant map in compact homogeneous space is a diffeomorphism

I don't see why an equivariant $G$-map $f: M \rightarrow M$, where $M$ is a compact homogeneous space, is necessarily a difeomorphism. Any idea?
Jotabeta's user avatar
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2 votes
1 answer
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Equivariant tubular neighborhood of an exceptional orbit of a circle action

A pseudofree $S^1$-action on a sphere $S^{2k-1}$ is a smooth $S^1$-action which is free except for finitely many exceptional orbits whose isotropy types $\Bbb Z_{a_1},\dots,\Bbb Z_{a_n}$ have pairwise ...
user302934's user avatar
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1 vote
1 answer
230 views

G-equivariance at the Lie algebra level

Let $\vec{x}, \vec{p} \in \mathbb{R}^3$ be coordinates on 6D phase space. Consider the lift of the standard action of SO(3): $$\psi_O(\vec{x}, \vec{p}) = (O\cdot\vec{x}, O\cdot\vec{p}) $$ This action ...
Mr Lolo's user avatar
  • 443
3 votes
1 answer
109 views

Orientation-reversing involution on $S^3$, commuting with a circle action? [closed]

Consider $S^1$ as a subset of $\mathbb{R}^2\cong\mathbb{C}$, and $S^3$ as a subset of $\mathbb{R}^4\cong\mathbb{C}^2$, and define an action of $S^1$ on $S^3$ by $z\cdot(w_1,w_2):=(zw_1,zw_2)$. Does ...
user919420's user avatar
6 votes
0 answers
193 views

Equivariant line bundle-divisor correspondence?

In reading about equivariant bundles, I've become a bit confused about how the usual line bundle-divisor correspondence $c_1:\text{Pic}(X)\xrightarrow{\cong}A^1(X)$ works in the equivariant setting. ...
Michael Mueller's user avatar
2 votes
0 answers
131 views

Induced representations built on sections of an associated vector bundle. Questions on notations

Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group: $$ {\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
Michael_1812's user avatar
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3 votes
1 answer
157 views

Equivariant Diffeomorphism $S^2 \times S^3$ to itself with respect to the following $\mathbb{Z}_4$ action

Let $\mathbb{Z}_4$ be the cyclic group generated by $(R,j)$ where $R \in$ SO$(3)$ is the rotation matrix $R = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{...
Tuo's user avatar
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2 votes
0 answers
61 views

Strange notation of fibre product in an article of Cattaneo, Felder and Tomassini.

I'm doing my bachelor thesis on explaining an article of Cattaneo, Felder and Tomassini, and here is the link to the article: https://arxiv.org/pdf/math/0012228.pdf. I'm currently working on the page ...
Abel-Henri-Guillaume Milor's user avatar
0 votes
1 answer
211 views

Equivariant rank theorem.

I was reading this post that explains why $$\phi :O_{n}\times H\rightarrow GL_{n}(\mathbb{R})$$ $$\phi(B,A)=BA$$ is a diffeomorphism. Here $O_{n}$ is the orthogonal group and $H$ is the group of all ...
roi_saumon's user avatar
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2 votes
1 answer
300 views

Pushforward of equivariant sheaf

I work over an algebraically closed field of characteristic zero. Let $G$ be an algebraic group, $X,Y$ varieties with $G$-actions, and $\phi:X\to Y$ a $G$-equivariant morphism. Let $\mathcal{F}$ be ...
freeRmodule's user avatar
  • 1,882
3 votes
0 answers
29 views

Why is the Tate map lax monoidal (just for abelian groups?)

Let $A$ be an abelian group with $G$ action, then I call the tate map $$(-)^{tG}: Ab^G \rightarrow Ab$$ $$ A \mapsto \operatorname{coker}(Nm:A_G \rightarrow A^G)$$ where $Nm:x \mapsto \sum_g gx$, is ...
Bryan Shih's user avatar
  • 9,618
3 votes
1 answer
270 views

Understanding implications of the equivariant Darboux-Weinstein theorem

I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here: The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example ...
Henry Shackleton's user avatar
0 votes
1 answer
68 views

Complex group representations as an enriched category?

In my lecture notes it says: ‘Complex representations of a given group G, together with intertwiners, form a category enriched over the complex numbers.’ Is it true that the category is enriched ...
Max Demirdilek's user avatar
0 votes
1 answer
79 views

long exact sequence of representations

Let $G$ be a group and $V_1,V_2 $ be $G$-representations. Are the ${\rm Ext}^i(V_1,V_2)$ $G$-representations as well? Once established this, suppose I have a short exact sequence $$0\to V_1\to V_2\...
Serser's user avatar
  • 189
2 votes
1 answer
86 views

What does it mean for right equivariant maps to be left translations?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
user avatar
0 votes
1 answer
19 views

For which values of a, b, c, d do $\theta_{a,b}$ and $\theta_{c,d}$ commute?

In Sets and Groups by Green a question 5 from the chapter 3 reads: Write $\theta_{a,b}$ for the map of the preceding exercise [which is $\theta(x)=ax+b=\theta_{a,b}$]. Prove $\theta_{a,b}\theta_{c,d}=...
user avatar
1 vote
1 answer
267 views

If pushforward by equivariant map of structure sheaf is structure sheaf and the space of sections isomorphic, are they isomorphic as G-modules?

Apologies for what may very well be a trivial question from a non AG person. Suppose I have a morphism of varieties $f: X\rightarrow Y$, with $Y$ affine, which is equivariant with respect to the ...
dz16's user avatar
  • 13
4 votes
1 answer
2k views

Does equivariance of the MLE require the function be invertible?

My statistics text states this theorem as if it works for any function $g$: Let $\tau = g(\theta)$ be a function of $\theta$. Let $\hat{\theta}_n$ be the MLE (Maximum Likelihood Estimator) of $\...
Joseph Garvin's user avatar
2 votes
1 answer
164 views

Taking homotopy fixed points preserves fibrations

I'm reading a paper where they have an appendix about homotopy fixed point sets of a $G$-space, and at some point they claim that if $f:X\to Y$ is a $G$-map that is an ordinary (non-equivariant) ...
Maxime Ramzi's user avatar
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