0
votes
1answer
40 views

relation between eigen values

Let $W$ be a finite subgroup of $GL(V)$ and hence it acts on $V$. Now consider the contra gradient action of $W$ on $V^*$. Now how to show that the eigen value of this action is the reciprocals of the ...
7
votes
1answer
58 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
3
votes
0answers
30 views

Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
2
votes
1answer
91 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
2
votes
1answer
128 views

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$.

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$. The element $m$ is $ \left( \begin{matrix} 2 & 1 \\ 0 ...
3
votes
2answers
66 views

action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
1
vote
1answer
53 views

Studying the action of $GL(V)$ on the vector space $V$

The statement I am trying to prove is the following. Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let $\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural ...
0
votes
1answer
52 views

Rank is an invariant group action

I have to show following things and have no idea what I have to do at b) a) $GL(n,K) \times K^{n \times m} \rightarrow K^{n \times m}: (g,A) \mapsto gA$ is a group action. b) The Rank is an ...
4
votes
2answers
147 views

If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?

Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$. Let $A$ be some ...