All Questions
11 questions
21
votes
1
answer
690
views
Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
- 45k
6
votes
1
answer
142
views
Stabilizers of multilinear forms
Let $\{e_1,\ldots, e_n\}$ be the standard basis of $\mathbb{C}^n$. Consider the $m$-multilinear form $$v=\sum_{i=1}^n e_i^{\otimes m}\in (\mathbb{C}^n)^{\otimes m}$$
and consider the action of $\text{...
- 5,039
5
votes
0
answers
321
views
Unitary representations of Tarski Monsters and other beasts
Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
- 647
3
votes
0
answers
816
views
Actions and representations of profinite groups
Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = \mathbb{Z}_p^{...
- 11.3k
2
votes
0
answers
280
views
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 ...
- 356
2
votes
0
answers
158
views
Finding invariant closed subspace which are also subgroups for the action of $\text{SL}(2, \Bbb Z)$ on $\Bbb R^n\times \Bbb R^n$
I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as
$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\cdot \big(v,\,w\big)\...
- 203
2
votes
0
answers
313
views
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 ...
- 21
1
vote
1
answer
511
views
Actions of $Z_n$ and actions of $Z_{n-1}$
I am playing with some questions concerning connections between
certain poset partitions and their linear extensions. This is not
my usual playground, I just happened to stumble upon something.
When ...
- 327
1
vote
0
answers
202
views
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 ...
- 13.6k
1
vote
0
answers
245
views
Find representation set of orbits when group acts on a set
Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....
- 337
1
vote
0
answers
337
views
Does a quotient group $G/N$ have a natural action on the regular representation of $G$?
Let $G$ be a group. I am happy to assume niceties such as finite and abelian, but perhaps it is not necessary to answer my question.
Consider the $|G|$-dimensional vector space $V$ (over some nice ...
- 607