# Symmetric power of a group representation

1. Aug 27, 2012

### anthony2005

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
$D$ of a group
$G$ ? “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 $V$ over a field and the tensor space $V^{\otimes r}$.
Thanks to the $GL\left(V\right)$ and $S_{r}$ groups, one is able to decompose the space $V^{\otimes r}$ in

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

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

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

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

$D_{g}^{\otimes r}\left(v_{1}\otimes...\otimes v_{r}\right):=D_{g}v_{1}\otimes...\otimes D_{g}v_{n}$

I also know that $D^{\otimes r}$ is in general reducible and I can decompose

$D_{g}^{\otimes r}=\bigoplus_{i}m_{i}\left(r\right)D_{g}^{i}$

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

$Sym^{r}D:G\rightarrow GL\left(Sym^{r}V\right)$

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

$Sym^{r}D_{g}=\bigoplus_{i}n_{i}\left(r\right)D_{g}^{i}$

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

Thank you very much.

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