# Thread: Group Action On Space of Multilinear Forms

1. ## Group Action On Space of Multilinear Forms

So, we know that if $\displaystyle \mathcal{V}$ is a finite dimensional [mat]F[/tex]-space and $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$ is the set of $\displaystyle n$-forms on $\displaystyle \mathcal{V}$ that $\displaystyle S_n$ acts on $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$ via the action $\displaystyle \left(\pi K\right)(x_1,\cdots,x_n)=K\left(x_{\pi(1)},\cdots, x_{\pi(n)}\right)$.

So, this might be a stupid question...but besides being able to state things easier (the symmetric forms are exactly the kernel of the action) how does this fact really help us? All the theorems I know which are useful and pertain to group actions apply for finite sets only.

2. One potential application: you could use this group action to show that any permutation $\displaystyle \sigma\in S_n$ is strictly even or odd (i.e. not both).

For a contradiction, you can suppose not: $\displaystyle \sigma$ is both even and odd. Try to see what happens when $\displaystyle \sigma$ acts on an alternating form $\displaystyle K$.

3. Originally Posted by roninpro
One potential application: you could use this group action to show that any permutation $\displaystyle \sigma\in S_n$ is strictly even or odd (i.e. not both).

For a contradiction, you can suppose not: $\displaystyle \sigma$ is both even and odd. Try to see what happens when $\displaystyle \sigma$ acts on an alternating form $\displaystyle K$.
That's a possibility, except that one defines alternating in terms of evenness and oddness and I feel as though it would be tough to divorce it from that.

Also, I am, not that what you said isn't a good idea, more concerned with traits learned about $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$

4. Originally Posted by Drexel28
So, we know that if $\displaystyle \mathcal{V}$ is a finite dimensional [mat]F[/tex]-space and $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$ is the set of $\displaystyle n$-forms on $\displaystyle \mathcal{V}$ that $\displaystyle S_n$ acts on $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$ via the action $\displaystyle \left(\pi K\right)(x_1,\cdots,x_n)=K\left(x_{\pi(1)},\cdots, x_{\pi(n)}\right)$.

So, this might be a stupid question...but besides being able to state things easier (the symmetric forms are exactly the kernel of the action) how does this fact really help us? All the theorems I know which are useful and pertain to group actions apply for finite sets only.
The kernel of a group action is a subgroup of the group, not a subset of the set!

In this case, the kernel is trivial, because given any nontrivial permutation, you can find a form which it doesn't fix.

5. Originally Posted by Bruno J.
The kernel of a group action is a subgroup of the group, not a subset of the set!

In this case, the kernel is trivial, because given any nontrivial permutation, you can find a form which it doesn't fix.
I'm sorry. This was late last night. What I meant to say is that the symmetric forms are those whose isotropy subgroups are the entirety of $\displaystyle S_n$.