Questions tagged [group-actions]
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465
questions
5
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Homogeneous metric connections on 3-dimensional Lie groups
Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...
- 51
4
votes
1
answer
110
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Examples of discrete quantum group actions on commutative von Neumann algebras
Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ ...
- 319
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105
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A compact Lie group $G$ acting on a compact Lie group $K$ transitively. Is there a $C$ such that $d(gx,gy)\leq Cd(x,y)$?
Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $...
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Action of braid groups on regular trees
Question:
Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...
- 3,347
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1
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72
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Nonamenable p.m.p. action on a standard probability space
Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.
Is the action of $G$ always amenable?
(Amenable action, ...
- 113
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1
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141
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An analogue of Mostow-Palais equivariant embedding theorem for the group of conformal automorphisms of the 2-sphere
Is there a smooth embedding of $S^2$ into some Euclidean space that is equivariant with respect to a linear representation of $PSL(2,\mathbb C)$?
A counterexample to a more general question can be ...
- 28.1k
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1
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46
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Holomorphic cyclic action on smooth toric manifold extends to C^* action?
Let $Z_n$ be a homological trivial cyclic action on a smooth toric manifold compatible with the complex structure, the does it extends to a C^* action?
- 31
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103
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Integral mean value property
Let $V$ be the space of all continuous functions $f$ on the real line with $f(x)=\frac12\big(f(x-1)+f(x+1)\big)$.
It contains the space of periodic functions. The latter equals the space of ...
- 1,672
2
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0
answers
65
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Bialynicki-Birula decomposition for $\mathbb{G}_m$-actions on projective schemes
The classical BB-decomposition works for non-singular projective varieties. Here I want to consider projective schemes, in particular when the scheme is not reduced.
Let $\Bbbk=\mathbb{C}$. Let $X$ be ...
- 860
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75
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Additional symmetries in Theta-like function
cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function
Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows
$$ \...
- 531
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125
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S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group
I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...
- 263
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0
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49
views
Lifting paths along group quotients relative to a base
Suppose you have a map of topological spaces $X\to S$, an $S$-group $G\to S$ (i.e. a group object in $\mathrm{Top}_{/S}$), an action of $G$ on $X$ relative to $S$ which is free and properly ...
- 445
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143
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Groups acting on categories produce 2-cocycles
$\DeclareMathOperator\Hom{Hom}\newcommand\id{\mathrm{id}}\DeclareMathOperator\Aut{Aut}$Let $\mathcal{C}$ be a category (such that each hom sets are $\mathbb{C}$ linear spaces) and $G$ be a group. We ...
- 9,008
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1
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dichotomy in hyperbolic groups
Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...
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Semi-direct products and associated graded Lie algebras
Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...
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51
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How to prove that pseudo entropy and topological entropy are the same with only Markov inequality and continuity?
Let $(X,\rho)$ be a compact metric space and $f:X\to X$ a homeomorphism. We say $(x_1,\ldots,x_{n})\in X^n$ is a partial $n$ orbit if $f(x_i)=x_{i+1}$. Let $Sep_{\epsilon}(X,\rho_n)$ be the maximal ...
0
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Examples of amenable, Hausdorff, locally compact, second countable groups which are not discrete, not compact, and not abelian
I'm working on a problem that involves an amenable group acting on some set by bijections. Initially, I assumed the group was discrete and the set was countable, however I realized that the arguments ...
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159
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Character tables of semidirect products on Sage
I am trying to find the character table of a semidirect product of two group with Sage. If I try the following I get an error.
...
- 226
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votes
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364
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Group action on a condensed set and its orbit space
Let $X$ be a condensed set, and let $G$ be a (discrete) group. Suppose we have an action $G$ on $X$, which is a group morphism $a:G \rightarrow \mathrm{Aut}(X)$, where $\mathrm{Aut}(X)$ is the group ...
- 1,057
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32
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Lifting a group action to a Banach bundle
I have been searching the literature for results on lifting a group action from the base space of a Banach bundle (Important note: NOT Banach VECTOR bundle). The setting I am interested in weaker than ...
- 1,898
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66
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Methods for calculating (one-parameter subgroup) actions
For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...
- 639
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0
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278
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A possible invariant associated to a compact group
Let $G$ be a compact topological group with normalized Haar measure $\mu$.
Is there an effective isometric action of $G$ on some $\mathbb{R}^n$ such that the following map would be a non-irreducible ...
- 303
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votes
1
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82
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Do we have an equivariant version of integrability theorem of flat connections?
I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1:
Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a ...
- 8,557
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0
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151
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Why Lubin Tate character acts on torsion points of CM elliptic curve implies the group of torsion points is infinite?
Let $F$ be quadratic imaginary field, and $R_F$ be its ring of integers.
Let $E /\Bbb{Q} $ be an elliptic curve which has CM by $F$. Suppose $E$ has good reduction at $P$,which is prime ideal of $R_F$....
- 1,087
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28
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Is there a sofic approximation for the free group on two generators satisfying $\sigma_i (a)(j)=j+1$?
Consider $\{a,b\}$ the generators of the free group.
With a sofic approximation I mean a sequence $\{\sigma_i\}_{i=1}^{\infty}$, with $\sigma_i:F_2\to \text{Sym}(d_i)$ satisfying
$\lim_{i\to\infty}\...
0
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0
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95
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Example of two equivariant structures on the same coherent sheaf which do not differ by a grading shift
Suppose we have a variety $X$ with a $\mathbb{C}^*$-action. If $\mathcal F$ is a $\mathbb C^*$-equivariant coherent sheaf on $X$ and $m \in \mathbb Z$, we define the grading shift ${\mathcal F}\{m\}$ ...
0
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0
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102
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When are all $\mathbb C^*$-equivariant structures on the structure sheaf $\mathcal O$ grading shifts of the trivial equivariant structure?
Suppose we have a variety $X$ with a $\mathbb{C}^*$-action. If $\mathcal F$ is a $\mathbb C^*$-equivariant coherent sheaf on $X$ and $m \in \mathbb Z$, we define the grading shift ${\mathcal F}\{m\}$ ...
3
votes
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103
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Unitary in adjointable operators associated with equivariant Hilbert module
Consider the following fragment from the article "Tannaka–Krein duality for compact quantum
homogeneous spaces. I. General theory" by De Commer and Yamashita:
How exactly is $\mathcal{E}\...
- 185
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48
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Different invariants of group actions from isomorphic subgroups
Consider $D_8,$ the dihedral group of order $8$, acting on the unit square $X=[0,1]^2 \subseteq \mathbb{R}^2$ in the natural way– essentially take the unique linear extension of the action on the ...
- 21
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Not very transitive actions
Suppose $m$ is a positive integer.
I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-...
- 41.6k
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39
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What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?
$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...
- 401
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1
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132
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Generalization of $G/T \simeq G_\mathbb{C}/B$
Let $G$ be a compact Lie group and Let $G_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.
Consider $H$ to be a Lie subgroup of $G$ and denote ...
- 129
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46
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Relation between weight spaces of fixed loci of Hamiltonian $S^1$-actions
Consider an almost Kähler manifold $(M,\omega,I)$ with a $I$-(pseudo)holomorphic $\mathbb{C}^*$-action, whose $S^1$-part is Hamiltonian and the fixed locus
$F=M^{S^1}$ is compact. Then, it breaks $F=\...
- 1,537
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94
views
Exists $G$-equivariant embedding with faithful representation of $G$?
Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
- 471
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Is the orbit foliation of the Weyl chamber flow Riemannian?
$\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold
$$
M_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma.
$$
Let $A\subset \SL(...
- 265
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Reference for rigidity of higher rank action
I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...
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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 ...
- 605
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60
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A weakening of infinite Golomb rulers for group actions
If a group $G$ acts on a space $X$, then a Golomb ruler is a subset $A$ of $X$ such that $|gA\cap A|\le 1$ for all $g\in G\backslash\{e\}$.
I am interested in a weaker concept, let's call it a "...
- 2,313
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votes
1
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225
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Paradoxical decomposition modulo finite sets
Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there ...
- 2,313
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votes
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237
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Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?
Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=...
- 8,557
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1
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117
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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 ...
24
votes
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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 ...
- 11.7k
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0
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79
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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.
...
- 129
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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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64
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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 ...
- 4,003
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votes
1
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295
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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 ...
- 20.4k
1
vote
0
answers
27
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Group action in the vicinity of an orbit where the stabilizer jumps
Consider a manifold $M$ with the action of a Lie algebra $\mathfrak g$. Suppose that the action is free,
except for one orbit $O\subset M$ where the stabilizer is a nonzero Lie subalgebra ${\mathfrak ...
- 111
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votes
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166
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Algebraic groups acting on affine varieties with finite-dim orbits in the coordinate ring
Let $K$ be an algebraically closed field of characteristic zero, and $X$ be an affine $K$-variety (identify $X$ with its set of $K$-points). Let $G$ be group acting "abstractly" on $X$, by ...
- 185
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votes
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267
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Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely
Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
- 484
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votes
1
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185
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Group action on fibre functor
(I asked this question on mathstack here: https://math.stackexchange.com/questions/4413271/group-action-on-fibre-functor. After getting no response and being suggested in the comments to post it here, ...
