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Symmetric power of a group representation

  1. Aug 27, 2012 #1
    Hi all,

    I am a bit puzzeled about group invariants. The main problem is, what exactly is the decomposition in irreducibles of the symmetric power of an irreducible representation
    [itex]D[/itex] of a group
    [itex]G[/itex] ? “Lie online” computes this decomposition for simple Lie groups. I've been typing on google this question since yesterday but nothing came up.

    Here is the starting point.
    Consider a vector space [itex]V[/itex] over a field and the tensor space [itex]V^{\otimes r}[/itex].
    Thanks to the [itex]GL\left(V\right)[/itex] and [itex]S_{r}[/itex] groups, one is able to decompose the space [itex]V^{\otimes r}[/itex] in

    [itex]V^{\otimes r}=\bigoplus_{\lambda\dashv r}\left(V_{\lambda}\otimes U_{\lambda}\right)=\bigoplus_{\lambda\dashv r}\left(a_{\lambda}U_{\lambda}\right)=U_{\left\{ r\right\} }\oplus....\oplus U_{\left\{ 1^{r}\right\} } [/itex]

    where [itex]V_{\lambda}[/itex] is the irreducible representation of the partition [itex]\lambda[/itex] of [itex]S_{r}[/itex] ([itex]a_{\lambda}[/itex] its dimension), [itex]U_{\lambda}[/itex] the irreducible representation [itex]\lambda[/itex] of [itex]GL\left(V\right)[/itex] (the latter can be viewed also as a schur functor [itex]U_{\lambda}\equiv\mathbb{S}_{\lambda}V[/itex] ).

    Amongst them there are two particular vector spaces [itex]U_{\left\{ r\right\} }=Sym^{r}\left(V\right)[/itex] and [itex]U_{\left\{ 1^{r}\right\} }=\bigwedge^{r}\left(V\right)[/itex] , both spaces of irreducible representations of [itex]GL\left(V\right)[/itex] .

    Now, consider a group [itex]G[/itex] and an irreducible representation [itex]D:G\rightarrow GL\left(V\right)[/itex] , [itex]g\rightarrow D_{g}[/itex] . I know how one can naturally define a new representation on [itex]V^{\otimes r}[/itex] [itex]D^{\otimes r}:G\rightarrow GL\left(V^{\otimes r}\right) [/itex]

    [itex]D_{g}^{\otimes r}\left(v_{1}\otimes...\otimes v_{r}\right):=D_{g}v_{1}\otimes...\otimes D_{g}v_{n}[/itex]

    I also know that [itex]D^{\otimes r}[/itex] is in general reducible and I can decompose

    [itex]D_{g}^{\otimes r}=\bigoplus_{i}m_{i}\left(r\right)D_{g}^{i}[/itex]

    where the sum is other all irreducibles [itex]D^{i}[/itex] of [itex]G[/itex] and [itex]m_{i}[/itex] the relative multiplicity (and one of them should also be the initial D , for some i [itex]D_{i}\equiv D[/itex] ).
    Now, the symmetric power representation [itex]Sym^{r}D[/itex] should be a new representation of G on [itex]Sym^{r}V[/itex]

    [itex]Sym^{r}D:G\rightarrow GL\left(Sym^{r}V\right)[/itex]

    How is it defined? Is it a subrepresentation of [itex]D_{g}^{\otimes r}[/itex] ? I immagine that it's reducible

    [itex]Sym^{r}D_{g}=\bigoplus_{i}n_{i}\left(r\right)D_{g}^{i}[/itex]

    What's the connection between [itex]n_{i}[/itex] and [itex]m_{i}[/itex] ?
    I've read some time ago that if [itex]D_{g}^{1}[/itex] is the trivial representation of [itex]G[/itex] , then [itex]n_{1}\left(r\right)[/itex] is the number of invariants of [itex]G[/itex] , built from the representation [itex]G[/itex] . Is this terminology correct?

    Thank you very much.
     
  2. jcsd
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