Results 1 to 5 of 5

Thread: Group Action On Space of Multilinear Forms

  1. #1
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    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 $\displaystyle K$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by roninpro View Post
    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 $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Quote Originally Posted by Drexel28 View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Bruno J. View Post
    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 $\displaystyle S_n$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Group Action
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 3rd 2011, 02:20 PM
  2. Group action G on itself
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Oct 21st 2011, 03:40 PM
  3. group action
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Aug 26th 2010, 12:07 PM
  4. Group action
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Apr 15th 2010, 12:49 PM
  5. Group action
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 15th 2009, 12:58 PM

Search Tags


/mathhelpforum @mathhelpforum